Tap the Hammer icon in the lower left corner of the screen to open the Building and Decor Shops.
On the left side of the screen, you can browse the different sections: Animal Homes, Production Buildings, Trees and Bushes, and Decorations.
Tap and hold the Building or Decoration that you want to use. Drag it over to an empty spot on your Farm, then tap the checkmark to place it.
For some buildings, you will need to select a Farmhand to complete the construction.