You use the mouse cursor for most things.
Left Mouse Button to select.
Right Mouse Button to cancel selection.
You can also use shortcut keys to make things faster:
1 -> Select slot 1
2 -> Select slot 2
3 -> Select slot 3
4 -> Select slot 4
5 -> Select slot 5
6 -> Select slot 6
7 -> Select slot 7
8 -> Select slot 8
0 -> Deselect
R -> Upgrade selected turret automatically
E -> Toggle Destroy Mode