Your City will already have a few businesses in place when you begin to play which you can leave in place or move to another location.
First locate the area you wish to build on and then tap on the "Build" icon which is a hammer in the lower right corner.
You will next see a build menu where you can choose between businesses, decor, factories, or residential buildings. Tap on the category you wish to use.
In example above you will see the "Residences" Menu which is very similar to all other menus. Simply tap on the item you wish to purchase noting the cost in coins or city cash and how many XP you will earn for building it.
Once you choose your building you will be taken back to the main screen.
Tap on the area you wish to build then tap a second time to confirm. You will see a "Construction" message and after a few seconds your building will be in place.
NOTE - If you make a mistake just tap in the correct area. It will not place until you confirm and even if it does you can correct it.
To Move an item simply tap on it and you will see the option to rotate it or move it into inventory. To move an item simply drag it with your finger to its new location.