Solve 10/25/50 Puzzles
These all will come naturally through normal play. You're likely going to need to solve, at the very least, 100+ puzzles for the Fossil Finder list anyway, baring some lucky drops or finding genome codes from other players. Any mix of all three fossil types will contribute to the completion count.

Feelings Corner
This relates to the main story. Every time there is an e-mail, there will be a red number on the PC icon, situated in the top left bar. Most of these are just hit reply deals and require no further work. The rest involve taking pictures with your Space bar of certain shells and layouts. Your e-mail list in the app will show a completion percentage so you have an idea of how much farther to go.

General advice:
-For X shell of X color for X amount requests, just place all the needed shells on one surface, zoom in by clicking on the surface and take the image.
-For furniture requests, placement doesn't matter and you don't have to keep it after taking the image.
-For Scrapbook requests, you don't have to take special images for each slot, you can fill them up with all the same of any old image.
-Every major tool will be unlocked from email/story progression, so don't forget to make them a priority.

Science Corner
After replying to enough emails, around 20% completion, the Genome App will unlock in the shop. Buy it as soon as you can, as it's going to be your most valuable achievement hunting tool going forward. This achievement unlocks the first time you use the app to generate a fossil. It's not noted in-game, but codes can be copy/pasted from in/out of the game, to make it easier to enter/share codes.

3/4/5/6/7/8 Star
This is likely the meat of why you're here. Discovering a high star fossil will unlock all the lower star achievement before it if you didn't already have them unlocked (IE: 5Star will also unlock 3 & 4Star, etc.). Doing this legit requires taking notes of traits, a pile of luck and gathering genome codes from other players. It's possible to get high star fossils without codes or combinations from the puzzle solving games, but it's very RNG dependent. Actually pretty fun to do all on it's own, the range of color/body/attribute combinations is wide and the displaying can be addictive.

But you're here for achievements, so just follow this formula:
[22AEQEGB05,[[17AZQQRU09,[[22AG17I603,1CALHCYD01],[22AG17I603,[22AG17I603,1CALHCYD01]]]],[[22AG17I603,1CALHCYD01],[22AG17I603,[22AG17I603,1CALHCYD01]]]]]

Step by step version:
-Using names isn't required, but suggested for keeping track. Names can be changed by clicking Edit after selecting a fossil.

1. 22AG17I603 & 1CALHCYD01 = Ben 1
2. 22AG17I603 & 1CALHCYD01 again = Ben 2
3. 22AG17I603 & Ben 1 = Ben 3
4. Ben 2 & Ben 3 = Jim
5. 22AG17I603 & 1CALHCYD01 = Sam 1
6. 22AG17I603 & 1CALHCYD01 again = Sam 2
7. 22AG17I603 & Sam 1 = Sam 3
8. Sam 2 & Sam 3 = Holly
9. 17AZQQRU09 & Holly = Max
10. Jim & Max = Ace
11. 22AEQEGB05 & Ace = Your new 8* fossil.

-Credit to lapine#5662 in the official Fossil Corner discord for 22A EQE GB05.
-Credit to *Mochi*#1323 in the official Fossil Corner discord for 22A G17 I603.

If there is an interest in doing this legit, which I do recommend if you're enjoying the game, check out this guide by Moldy Bread. They explain the process excellently:

https://gtm.steamproxy.vip/sharedfiles/filedetails/?id=2633568047

Fossils FOUND!
After some story progression the Fossil Finders app will be unlocked. This consists of four challenges, each with their own reward (that you'll then have to buy....). All of them are for taking pictures of particular fossils in various body/color combinations and sometimes as a set. The achievement itself is for unlocking every shop item possible. There isn't really any straightforward way to guide this. It's all up to luck. With good rolls, picking through genome codes and a streak of less than annoying challenges, it can be cleared anytime between a few hours to several hours.

General advice:
-This may or may not have become obvious earlier in the game, but since it's not implicitly pointed out, if a fossil has more than one attribute, so long as it also has the ones needed for the challenge, it'll count. That is, challenge asks for 2Star, Green, Small Trilobite and you have one that has those three attributes and six more, its still covered the requirement.
-A single challenge can be rerolled every 10 minutes. Use this to your advantage to remove troublesome ones. The game has an annoying habit rerolling challenges into similar challenges with harder requirements.
-The otherwise not particularly useful Rock Tumbler can be used for trading in/out traits if it works correctly. If you don't mind backing up saves left and right, it's an option for brute forcing desired fusions.
-Having a healthy collection of all three fossil types will help provide combination fodder for challenges.
-It's worth your time to clear puzzles until you've got the cash to access all three walls of the work space and upgrade to the largest storage units. I would suggest making this a priority. Make use of your stash and pockets as well.