Welcome to the game. We were all new once.
SHORT ANSWER
You roll off for who gets first turn before the game starts. The first player does his command, move, psychic, shooting, charging, fighting, and morale phases. Then the second player does all those phases.
LONGER ANSWER
In most phases, the player whose turn it is is the one activating units and doing stuff. However, the other player will have opportunities to do stuff on your turn.
This is most obvious in the charge phase where the other player can choose to overwatch, heroically intervene, utilize the Defensible terrain special rule, etc., and in the fight phase when you and your opponent take turns activating units. (See the main rulebook's section on the Fight phase for more details.)
Also, some special abilities and stratagems can be activated on the other player's turn. For instance, the Auspex Scan stratagem lets you shoot with one of your units during your opponent's movement phase.
So you're basically correct, but there are caveats.