Let's Build a Zoo features 45 achievements, 12 of which require the two DLCs "Dinosaur Island" and "Aquarium Odyssey".

I've added additional info to the more involved ones, which should hopefully help you to complete all of them.

These are linked to the various progression paths you'll complete as you expand your zoo.

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Super Trader
Complete 10 trades with other zoos.


The world map contains several possible trades, where other zoos will offer you a pair of animals in exchange for a certain animal in a specific variant. Simply complete any 10 of these to get this achievement.
For mor info on obtaining the correct variant, check the achievement "Cartographer" below.

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Lion City
Complete the trade for the Lion in Singapore.


The trade for the lion is one of the last ones to be unlocked, meaning you'll have to complete most of your animal collection first.

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Cartographer
Map your first Genome.


For this one, you'll have to find all 10 variants of a single kind of animal, which are unlocked through breeding in the nursery building.
After selecting the animal you want to breed, the game will give you an overview of the possible variant combinations you can use. Each of these options have multiple potential outcomes for the variant the produced offspring will have. These often have different probabilities as denoted by the bar below, though these eventually even out somewhat as you actively breed each option.
Simply breed available pairings that have a chance for creating new variants. Also try to go for variants which enable new breeding pairs, giving you access to new variants. Expand your breeding capacity and try to focus on completing the genome of one animal at a time, as that makes it much easier to sustain a varied population and focus your breeding attempts.

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Map Master
Map 56 standard animal Genomes.


Contrary to the achievement's description, you'll have to unlock all 10 variants for all animal types available in the base game, putting the figure at 67 as of me writing this guide.
This obviously requires you to complete most of the game in order to gain access to all animals. However, you should try to work away at this steadily throughout your playthrough, as it takes quite a while and isn't very engaging.
For specific more information on the breeding process required for this, take a look above at the achievement "Cartographer".

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Researcher
Research 20 things.


After unlocking the research hub, simply use your research points to unlock any 20 of the available options.

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Superior Researcher
Research 233 things.


This requires you to unlock most of the available options in the research hub. By now there's quite a few more research options available.

Throughout the game, different characters will ask you to complete certain objectives. The following achievements simply require you to complete some of these quest lines.

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Captain Cola
Convince Captain Cola to run a promotion at your zoo.


Captain Cola will approach you rather early on, asking you to place various items related to his brand in your zoo multiple times.

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Route Planner
Unlock 5 bus routes.


Steve the transport planner asks you to reach certain visitor numbers on multiple occasions, unlocking each of the 4 additional bus routes as the respective reward.

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He Sells Sanctuary
Complete the Goth quests.


Andrew the Goth requests you to extend the opening hours of your zoo multiple times. To do this, simply research the different kinds of lamps in the research hub.

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Sky Writer
Get a flying visit!


Rather late in the game, you'll get a single quest from Russel Case asking you to breed a flamingo. As your reward, he'll start promoting your zoo, flying across it with his plane.

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Monster Matinee
Complete all Kaiju Cosplay quests.


Tomoyuki approaches you early on, giving you multiple quests to deliver them certain monster-like animals.

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New Zoo
Unlock a second zoo on the world map.


Kathleen the Investor will give you a lot of tasks, guiding you through the game and towards expanding your zoo further and further. After completing all of them, you'll be allowed to buy a second zoo by clicking on the green "Transport" icon in the upper left corner of the world map. From there, you can simply select the dot in North America and buy the second zoo.

For these achievements, you'll essentially have to expand and develop your zoo, meeting various criteria related to its success.

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Decorator
Get a Deco Rating of 100%


This one simply requires you to place a lot of decorations throughout your zoo, with the respective heat map showing you how well you're doing with this.
This is easier to achieve as you start out, when you only have a small zoo with few tiles of land to cover.

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Transport Mogul
Own at least 10 buses.


You should naturally unlock this sooner or later. Simply have 10 busses of any kind, spread across the different routes in any way you like

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Cash King
Earn at least $50000 in a single day.


As you expand your zoo, it will attract more and more people willing to spend more and more for their tickets. You'll get this naturally as you play through the game.

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Ticket Master
Sell a ticket for at least 100 dollars.


You can at any point use the entrance gate of your zoo to set the ticket price.
However, this achievement will only pop if at least 1 visitor actually buys a ticket for this price, meaning you can't just get it straight away. As your zoo grows larger, people will be willing to spend more to enter it, eventually allowing you to successfully set the price to 100 bucks as required.

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Land Owner
Buy every plot of land in the starting zoo.


As it says, simply buy all the available land around the starting area. This will mostly require a fair bit of grinding towards the end of the game, as each additional piece of land increases the price for the next one, which gets to some rather ridiculous numbers towards the end.

Let's Build a Zoo features a Morality system, letting you develop your zoo in a good or evil way. The following achievements either require you to follow the good path or facilitate it by boosting your good score.

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Snitch
Send a member of the black market to prison.


During the game you regularly encounter an event in which you're offered money in exchange for certain animals. Simply use the option it gives you for reporting the dealer to the police.

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Green Machine
Build your first Wind Turbine.


The wind turbine can be unlocked from the research hub and requires at least 5 good points to be build.

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The Saint
Have 200 good points.


You need to reach the highest number of good points possible to unlock this achievement.
Clicking on the Morality-meter at the top of the screen will bring up an overview of your current morality status. From there, selecting any of the categories will show you the different options for gaining good (and evil) points. You'll have to max out each of these options.

Maxing out the bars seems to require you to not have any evil buildings/items in your zoo which negatively impact the respective effects at all.

The Morality system featured in the game allows you to develop your zoo in different ways. These next achievements either need you to pursue the evil path or facilitate following it by boosting your evil score.

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Bad Hot Dog
Sell a Hotdog at the worst possible quality.


Simply build a hot dog stand, select it and click the "Manage Shop" button. Next, move the "Adjust Recipe: Meat" slider in the middle of the window all the way to the left, towards "Eyes and Tails". You'll probably also have to adjust the "Adjust: Seasoning" slider in the top right so it reads "0%".
If needed, adjust the price downwards on the window's left side to lure people in. Otherwise just wait for someone to buy a hot dog.

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Broken Teeth
Break some teeth.


This achievement is part of a quest, in which Dr. Driller the dentists asks you to sell 500 portions of unpopped popcorn to be rewarded with some evil points.
To fulfill the requirement, build a popcorn stall and select it, clicking the "Manage Shop" button that comes up. Then, move the "Adjust Recipe: Popcorn" slider in the center of the window all the way to the left, towards "Unpopped". You may have to adjust the "Adjust: Salt" slider in the top right as well, so it reads "0%".
Think about reducing the price on the left side of the window to get people to actually buy your junk, then simply wait until you've sold enough to complete the quest.

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Protest
Have a visit from a protestor.


Protestors will arrive if any of your animals are not properly being cared for, e.g., when they don't have enough water or enrichment.

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Dealer
Sell 10 animals to the black market.


You'll regularly encounter an event in which you're offered money in exchange for certain animals. Simply accept this offer 10 times.

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Tastes Like Bacon
Build a bacon factory.


The bacon factory can be unlocked from the research hub and requires at least 10 evil points to be build.

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Factory Boss
Have at least 6 unique types of evil factory operating in your zoo.


There are currently 13 factories in the game, each requiring between 5 and 100 evil points to build. Simply construct 6 different ones out of them all.

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Criminal Mastermind
Have 200 evil points.


For this one, you need to reach the highest number of evil points possible.
Clicking on the Morality-meter at the top of the screen will bring up an overview of your current morality status. From there, clicking any of the categories will show you the different options for gaining evil (and good) points. You'll have to max out each of these options.

Filling up the bars seemingly requires you to remove all buildings/items that counteract the respective effect in any way. To max out your emissions in particular, it seems like you need to remove every single plant, even those that don’t display that they’re reducing CO2.

These achievements have quite a variety of requirements and don't really fit in any category. Most of them are rather basic through.

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Have a baby!
One of your animals must give birth


You should get this basically instantly by simply having two animals of opposite gender exist in the same enclosure.

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Big Birther
Your breeding program must produce 100 babies (Nursery births only).


This one requires the nursery, which can be unlocked in the research hub and enables you to manually breed two animals. Simply do this 100 times, like you'll do multiple times over for the "Cartographer" and "Map Master" achievements.

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Hybrid Moments
Create your first hybrid animal.


For this achievement you'll need the CRISPR Splicer unlocked early on via an event.
Just construct the building. Then select it, choose the "Select DNA Pair" option and choose any two animals of yours from different species to splice. Then simply wait for the splicing process to be completed.
Animals are only available for splicing once you've unlocked at least 5 of their variants. For more info on how to do this, check the "Cartographer" achievement above.

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Trampoline
Buy a trampoline for your zoo.


The trampoline is one of the earliest unlockable enrichment items. Simply unlock it in the research hub and place one in any enclosure.

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Gonky
Employ a Gonky mascot.


Mascots are unlocked through the research hub, at which point you can hire them by clicking on the entrance gate of your zoo and using the "Park Staff" button. There, simply open a mascot job posting and wait for applications. Each mascot will by of a random type, so just wait until a Gonky shows up and hire them.

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Fired Up
Fire an employee.


For this achievement, select any employee by clicking on them or through the "Park Staff" menu accessible via the entrance gate of your zoo. Then simply select the "Fire" option and confirm that you want to let them go.

The game's first DLC features some new achievements, most of which basically require you to create a zoo on dinosaur island and complete the DLC's campaign.
You can acquire the dinosaur zoo at any point in the game for free, by entering the world map and clicking on the green "Transport" icon in the upper left corner. From there, simply select the dot on dinosaur island (The island in the middle of the Atlantic featuring a volcano) and buy the zoo.

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Dinosaur Rebirth
Clone your first dinosaur.


This is the first thing you'll do when entering the dinosaur island zoo.
Simply select the CRISPR Splicer already built in the zoo, click the "Clone Animal" button and select the available dinosaur. Then just wait for the cloning process to be completed.

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A Mammoth Discovery
Complete the dig for the Mammoth.


In this DLC, new species are discovered by investing in dig sites found on the world map. The mammoth dig should become available rather early on.

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Dino-Mite
Discover all 51 Dinosaurs/Animals.


There are 51 dinosaurs and prehistoric animals in this DLC, with this achievement requiring you to unlock all of them. Simply complete all digs on the world map and you'll eventually get them all.
None of the base game animals count towards it.

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Pooped Out
Build a Bio-waste Storage in Dinosaur Island.


The bio-waste storage is unlocked through the research hub.
Contrary to the achievement's description, I got it by adding the building to a regular zoo.

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Bernie And Friends
Employ a Bernie mascot.


Bernie is the only kind of mascot available in this DLC, meaning this will unlock once you hire the first mascot.
Mascots are unlocked through the research hub, which sees them become available for hire. Just select the entrance gate of your zoo and click the "Park Staff" button, which allows you to a mascot job posting for mascots. Then simply wait for applications and accept any one of them.

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Death Metal Death
Complete all Critical Choices for King Sapphire.


As your playthrough of the DLC progresses, you'll receive multiple events involving King Sapphire, who wants to enter your enclosures. You can either accept or deny his requests and will unlock this achievement after dealing with the last one either way.

The second DLC for Let's Build a Zoo comes with its own set of achievements, most of which you should unlock just by playing through its campaign.
The aquarium zoo can be entered for free at any point in the game. Simply open the world map and click on the green "Transport" icon in the upper left corner, then select the dot on Madagascar (The island at the lower right of Africa) and buy the new zoo.

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First Catch
Get first fish from either boat.


This will be the very first thing you'll do when starting the DLC. Simply open the world map, select either of the two boats and buy at least one of the creatures they offer.

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Whale Done
Get a blue whale.


For this one just progress through the game and collect new animals. You'll receive a quest for getting the blue whale quite early on, though (at least for me) it'll take quite a while for it to become available from the ships.
Take a look at "Seas the Day!" below for some basic tips on how to get new sea creatures.

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Seas the Day!
Collect 50 Aquatic Animals.


There are 50 water creatures featured in the DLC, meaning this achievement requires you to unlock all of them. Make sure to upgrade your ships using the research hub, as this enables them to actually find all creatures. There's also the FFFI agent, a guest who visits from time to time, offering you one of a selection of new creatures when you interact with him.
None of the base game animals count towards this achievement.

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Sea Monster
Create your first marine hybrid.


As with the similar main game achievement, you'll need the CRISPR Splicer building for this one, though you'll be able to construct it right away in this DLC. As the DLC also lacks animal variants, you'll just need to have had at least two different sea creatures in your zoo before you're able to make use of the CRISPR Splicer.
Once construction is done, select the building, choose the "Select DNA Pair" option and choose any two sea creatures from different species. Then simply wait for the splicing process to be completed.

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Pearlfect
Build a Jeweller.


The Jeweller is unlocked through the research hub.

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DGORSD
Hire 6 different types of mascots.


This one requires you to hire every type of mascot available in Aquarium Odyssey. The achievement is based on completing the mascot employee collection (accessible through the management office), so you don't have to employ all 6 of them at the same time - having hired each mascot type at some prior point is sufficient.
Note that there are two types of shark mascot with little visual difference.

I hope I could help you to fully complete the game. Feel free to use the comments to tell me if I made any mistakes or wasn't clear in my explanations at any point.

I'd also very much appreciate it if you rated this guide and/or gave it an award in case it did help you.