The game itself is actually set after the events of the movie, so you're unlikely to uncover too many spoilers in it, and consists of a series of space battles, where you control the Enterprise against oncoming waves of Klingon, Cardassian and Romulan enemies.
Enemies float in from the top of the screen, leaving you to dodge in and out of their fire and throw out a few shots of your own too. If you're one of those un-coordinated sorts who can't cope with the concept of moving and firing at the same time, you can always set the game into auto-fire mode and let the AI take care of the shooting for you. Whether auto-fire is on or off, the close range phasers on your ship will fire automatically anyway, meaning all you have to take care of are Photon Torpedoes (activated with a press of 5 or the Action button) and any special weapons that you grab along the way (which can be utilised by pressing 0). Special Weapons and Power-Ups/Multipliers will float around in space, and if you get close enough, they'll be pulled in by your tractor beam.