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Disclaimer: This guide is written during Civilization 6's launch week in October 2016. Future updates and patches may make some of these tricks worthless. I won't be able to retry the Achievement tricks after every game update. Therefore I will not update this guide so it might be possible that the tricks you are reading here are not working anymore if you read this months after the initial release of the base game.
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Hi Folks,
here are some very cheap ways that I found to get a ton of achievements very fast in Civilization 6. You can call these methods cheating. Or you can think "it's just another strategy, albeit not IN the game but AGAINST the game". ;-) It is up to you if you want to boost the achievements using the following instructions. It takes the value out of the achievements, especially for the Deity Difficulty one. But this only matters if Achievements have any value for you to begin with ... if you just need them to quench your thirst for completion, read on :-)

Step 1:
The Difficulty Achievements

The Difficulty Achievements stack. This means that if you win a game f.ex. on Prince Difficulty, all the lower difficulty achievement will unlock as well. So in conclusion, only one win on Deity Difficulty is needed to get ALL of the Difficulty Achievements! Since this is a guide to get the achievements very quickly, we need a fast method to get a Deity win.

And here is how you can do this: Just start a 1 on 1 game against any Civilization whatsoever, using any map type etc. but go into Advanced Settings and
1) deactivate every win condition except Score
2) Set Max Turns to 1

As soon as the game starts, put down your first city and end the turn - you will instantly win the game and the achievements are yours!

(Note: If for some reason this doesn't work, play as Rome. They always start with a building right away so when you put down your first city it will have a building in it, which garantuees that your score is higher than your opponent's score)

This will give you all seven Difficuly Achievements at once, plus the Achievement for Winning a game on the chosen Map Type plus the Achievement for Winning with the Chosen Civilization plus the Achievement for Victory in the chosen Starting Era. So this method nets you at least 10 achievements in a matter of seconds!

Step 2:
The "Win with a certain Civilization" Achievements + "Map Type" Achievements + "Starting Era" Achievements

Now what you can do is repeat Step 1 over and over again but every time you choose another Civlization, another Map Type and another Starting Era. By doing so, you will unlock a huge chunk of Civilization 6's base game achievements in a matter of minutes. (Loading times not included ;-) )

Step 3:
Religious Victory

This takes more time than the first step but it is still a quick way to get an achievement. What this step does is basically guaranteering you a Religious Victory, so this method can be used to get the Nirvana Achievement and the We are the Champions Achievement, although the latter takes some planning and time.
Still, knowing that you can be sure to win a Religious Victory is very helpful for those two achievements ... so, how do you do this?
Very simple: Just play 1 on 1 against Kongo! Because here is the thing: One of Kongos "traits" is that they are unable to found a Religion! Therefore, they can never win a Religious Victory. So create a New Game, set the Win Condition to Religious Victory only and play against only one AI, which of course needs to be Kongo! Then just go on and win the game.

Step 4:
Achievements that are generally easy to get

Now here are some achievements which you will almost certainly get at one point by just playing normally. But if you are going specifically for them, they are only a matter of minutes.

1) Naming Rights: You just need to rename a unit. You can rename a unit by clicking a button in the units GUI, however it only becomes available after the unit received two promotions (level-ups). Just start a game with barbarians and use the starting warrior unit to smack them around until you get enough to XP to receive two promotions, then rename the unit and ... voilà! Achievement get!

2) A Revolution Without Dancing: You just need to change your government for this one. You start with the Chiefdom government. Click on the Tech Tree for Civics and choose on of the early governments (Autocracy for example) and just play until you unlock it, then change the government. If you play on the quickest speed setting, this should not take long at all.

3) Escort Service: You always start with a Warrior unit. So start a new game, build a city and immediately build a second Warrior unit. As soon as it is finished, put them together by clicking the "Linking" Icon in the units GUI and you will get the achievement.
Comment by user Lupal Fillyus: This isn't about forming a corps/army, it's about a formation of combat-non-combat units. Just link your Warrior with your Settler
To do so, have the Settler and the Warrior unit nexrt to each other and click the "Linking" Icon in the units GUI.

4) Land Party: You just need to start a Multiplayer game. Easy peasy!

There you have it. If you have done everything in this guide, you will have over 50 of the 100 of Civilization 6's achievements in easily an evening. Most of the achievements that are left now are not very hard nor very time consuming, but they demand very specific things to happen and often need you to finish/win a game under certain conditions. Unless you specifically go for them, you probably will not get them by chance during normal play. But these are not as quick to get as the ones mentioned here and since this is a guide about QUICK achievements, this is were it ends. Thanks for reading! :-)