All production buildings become available for purchase as your farm level increases. You can see the required level for the building in the Shop (the basket button in the lower left corner) in the Buildings tab.
To build an unlocked building, tap on it in the Shop and place it anywhere on the farm where the arrow markers are highlighted in green. If the arrows are highlighted in red, there is an obstacle preventing you from placing the building.
In most cases, you can only buy and construct one of each building type, but there are exceptions: Feed Mills (2 buildings) and Foundries (3 buildings).
Some buildings require special resources to build: planks, nails, bricks or screws. You can get them from chests or daily rewards, by completing tasks on the Beach, and by buying from other players in the Global Market.