The button in the top right corner toggles the keyboard.
Under that are the three layout buttons, corresponding to the three main screens in Colonization:
If the right layout isn’t shown, tap one of the buttons. Tap another time to hide the layout.
On the top left are three buttons that are useful on any screen: Info (F1), Enter, and Unload (u).
On the Map screen, the movement controls are on the bottom right:
The compass rose is an 8-way D-pad. Tapping the centre of it centres on the current unit.
The “zzz” button lets the current unit wait and activates the next unit.
The footsteps button orders the unit to go to a colony (or Europe).
The “Move / View” button toggles between moving units and viewing map tiles.
The orders you can give to a unit (Build, Fortify, etc) are on the bottom left.
Any other commands can be accessed from the drop-down menus, since they’re not used as often. (Please comment if you think an important command is missing from the UI.)
On the Colony and Europe screens, you can move part of a cargo. Click “Split cargo” and drag cargo between the warehouse and transport. The dialog will pop up. Fill in the amount you want to move by opening the keyboard (top right corner).