WARNING!!!

The game initially tells you to build a Coop but this will only put you into danger. I recommend building a Silo first before even considering getting any of the buildings below.

You can't skip upgrades. You have to build each building in order.

Can house up to 4 animals.
Unlocks cows.
Dimensions: 4 x 7
(All upgrades have the same dimensions)

Can house up to 8 animals.
Unlocks goats.

Requirements: Barn

Can house up to 12 animals.
Unlocks sheep and pigs.

Requirements: Big Barn

Can house up to 4 animals.
Unlocks chickens.

Dimensions: 3 x 6
(All upgrades have the same dimensions)

Can house up to 8 animals.

Unlocks ducks.
Provides incubator for hatching eggs in livestock.

Requirements: Coop

Can house up to 12 animals.

Unlocks rabbits.

Comes with an auto-feed system.

Requirements: Big Coop.

Allows you to collect hay from grass.
The maximum amount of hay per Silo is 240.

Dimensions: 3 x 3 

Creates an area where you can access water.
Ideally used for the watering can.
Dimensions: 3 x 3 

Horse is included.

Dimensions: 2 x 4

Can hold up to 20 slimes.

Dimensions: 6 x 11

To help you plan your farm, you can use a virtual planner.

https://stardew.info/planner/

Disclaimer: I did not create the planner. I give full credit to whoever created it.