These achievements focus on campaign.

It Was Supposed to Be an Easy Job...
Enter campaign mode. You'll get it after the end of first cutscene/intro.


First Day at Work
Complete first level.


Tea at the Garden Party
Reach 3rd floor.


Song of Ice and Snow
Reach 5th floor.


Manor at the Top
Reach 7th floor.


Halfway There!
Finish 15 levels.


Dungeons and Wagons
Reach 9th floor.


Who You Gonna Call?
Reach roof floor (highest floor achievable by elevator).


5% of Players Do This
Go to the 30th level. You have to take elevator to highest floor. Then climb two ladders up.

All-Star Player
Accumulate 90 stars. Basically get 3 stars for every level.

All of these achievements require specific map to unlock. I marked levels with numbers from party mode and campaign mode.

Legend:
PM = party mode
C = campaign

Fishing Time
In levels [2.1 and 2.2(PM)/6 and 7 (C)] there are water areas. There are carpets and buckets swimming in pool/river. You just have to fish one of them out of water.

Doggy, No!
In level [4-4(PM)/19 (C)] there is a dog. It often blocks your way. You have to stumble over it.

The Floor Is Lava!
In level [5-3(PM)/23 (C)] there are lava pools. You have to walk into lava.


Geronimooo!
In level [6-1(PM)/26 (C)] there are bridges. You have to fall off of one of them.


Swimming Time
In levels [2.1 and 2.2(PM)/6 and 7 (C)] there are water areas. You have to be thrown into water by other player.

I Can't See It!
While holding a blueprint press either [Q], [E] on keyboard or any of bumpers or triggers on controller.

Expert Botcher
Use wrong on paint on wall. You can see what paint is right on blueprint.


Breach, Clear!
Hold [F] (keyboard), X (Xbox controller), ⬜ (PS controller) next door. Front doors don't count!

Ain't No Shakin' Without Breakin'
Put garbage into bucket that already has something in it other than rubbish (like glue).


Three-Point Field Goal
Simply pick up another player and throw him out.


Catcher
Catch something midair. Can be easily done while playing co-op.


Catch 'Em All!
Earn 75 stars in campaign mode to unlock all characters.


Expert Doom Architect
You have to slip on something while holding glue/paint bucket/bucket full of something. Then make another player slip on your stain.

It's a Trap!
Delivery guy always comes one way in current map. Just throw/slip with glue/paint bucket/bucket full of something to make a stain on his way and let him slip on it.

Creative Architect
Destroy wrong wall. You can see how walls should be right on blueprint.


Good Times
You have to press [Space] (keyboard), A (Xbox controller), X (PS controller) while midair. You have to be thrown into air by other player to do this. Lava won't work.

Leave Me Alone!
First you have to be hold by other player. Then try to wiggle out. On controller try to move left-right with analog stick. On keyboard try to spam [A], [D] keys respectively.

Got Ya!
Catch another player midair.


You Shall Not Pass
This achievement requires you to install a door in wrong place. You can kick doors out or destroy walls that are adjacent to doors. Then either demolish walls to make one space for doors between wall or build 2 walls with space in the middle.
Best Party Ever
Complete all maps in Party mode on 3 stars. Note that you unlock levels in Party mode by completing them in campaign!

Couch Co-Op
You have to play as 4 players and all of them have to hold couch in one moment.


Platinum
Earn all of the achievements possible.

NOTE: Mode was added during the update at 23.09.2020 and requires completing campaign mode beforehand (You can't start mode without completing campaign mode first).

Time Conquered
You have to earn at least 1 hourglass in each level in Time Attack mode.


Veni, Vidi, Excellui
You have to earn 3 hourglasses in each level (complete with specified reserve of time) and make "no flop". Flop include actions such as: painting walls with wrong paint; using wrong panels/carpets; making a stain; putting rubbish on floor etc.

Here are achievements that were added alongside Garden Party DLC (14.04.2021).
I decided to categorize them based on what is needed to get them.

I just want to add that the DLC have two achievements that require 4 players, but fear not. When on PC you can turn on "Split keyboard" option in Options -> Gameplay, which enables you to earn them alone. Here is the button layout:

You don't need to have any DLC in order to unlock these.

I like boats
You have to throw 50 objects into water intentionally.


Clear all the oceans
Just pick 51 objects from water. You can unlock this while trying to get "I like boats" achievement. This is also easily achievable in new DLC as there are many water areas.

Who’ll let the dogs out?
On level 7 you can pick the dog up and press shift/B/circle to pet it :).
*Note* There is no possibility in petting dog from base game unfortunately.

Diglets
You have to pick up a shovel and move next to the other player. While other player is marked with a white outline you can press action key (F/X/square) to pick him up with a shovel.
Over the hedge!
After picking up a player with a shovel you have to press action key again to throw him away.

Movin in!
I think first map you can do it on is the 8th one. Just put everything in wrong places.


Secario
This comes naturally. I got it on the last level of first part of DLC. Just cut 100 grass tiles with lawn mower.

Solid Hose
This is achieved automatically during progression. There will be a level where you have to water every plant with a garden hose. I got it on level 8 of the first part of DLC.

Burglary to the fullest
This achievement also comes naturally. Racoons will notoriously steal objects from you and throw them into water.

Grass artist
This is achievable in the last level. You have to cut the grass with bare hands, lawn mower and pruning scissors.

A friend or foe?
This one is quite easy. You have to let the dog plow out grass debris (after you remove top layer of grass there will be a second layer). You can get it on 7th level.

Movin out!
There must be 4 characters at level, each carrying 1 furniture piece individually. Easly obtainable on level 8.

Ouchy
Get stinged by cactus 25 times. Pretty self-explanatory.


Traveler
You have to travel through 5 lava tiles in one map. I've done it on 27th level as there was an opportunity to use them to make travel more efficient.

Full rock house
You have to hit a rock with a pickaxe till all tiles around it have material on ground. Just keep hitting it. I got mine on 21st level as it is first map with rock.

Inside job
You have to "banish" the mole. It's a simple achievement. First mole fan be met at level 20.

No pain without gain
Clean 25 Mole Ground Piles. Got it normally during playthrough, at last level of 2nd DLC.

Lavash
Throw 25 objects into lava. I farmed it on level 24.
*Note* Throwing objects in base game levels doesn't seem to work.

It’s outta control!
You have to put paint/rubbish bin near the hose and turn it on max power hoping it will turn it over (10 times ;-;)

Pool party
You have to throw all throwable objects (plants, tools, buckets) into a swimming pool. I did it on 22nd level. You don't have to have blueprint/furniture/tools from bushes in the swimming pool.
Autumn at finest
Scatter all leafs to the fullest. Just walk through pile of leafs several times. It's easy to obtain on map with only one leaf pile (for example level 34).

World Cup
Simply push ball into a goal. Can be done on any map with goal/s and ball/s (like level 34).

Ghost Truster
Let toxic ghost destroy 10 objects on the level. I've done it on level 35 just trying to complete it...

Pusher 3000
Push toxic ghost outside the level. You can hold your dog and go towards the ghost till he flies outside screen.

Beware the dog
Simply walk towards the ghost with a dog in your hand.


Michelangelo Buonarroti
Sculpt a sculpture in 10 seconds or less. Easy to do with second player/controller on level 37.

Forest planner
Take out tree without leafs, dig up the hole and put it in.


For Three!
Throw a plant into a box. You won't be able to complete later levels without earning this one.

Rock band
Pick cobble with 4 characters.


GhostBuster
Don't let toxic ghost destroy anything on the level. Yeah, good luck ;-;.
I've done it with my friend, this might help someone:
https://www.youtube.com/watch?v=xaTgKncWPf4