How a turn works
Each day every army takes one turn. On your turn, every unit can move and act once, and you can build at each empty factory, airport and port you own. Press End turn when you are done.
Tap a unit to see where it can go (blue tiles). Tap a tile to move there, then pick an order: Fire, Capture, Wait, Load and so on.
Funds
At the start of your turn you earn funds for every city, factory, airport, port and HQ you own (1,000 each in a normal game). Comm towers and labs pay nothing but have other uses.
Winning
Capture the enemy HQ, or destroy every one of their units. Some games also end at a day limit or a number of captured buildings; the game's settings say so.