Learn Danish: Vi kører gennem skoven. - We drive through the forest.

Gennem is a preposition, which means it must be followed directly by an object (in this case, skoven). Igennem is an adverb, so you would use it if there is no object following it, or if you place it after the object (like in the alternative phrasing: vi kører skoven igennem).

Can kører mean we are just passengers, or does it strictly mean we are driving?

It can mean both! While at køre translates to "to drive," Danes use it generally for traveling by any wheeled vehicle (car, bus, train, or bike). You don't have to be the one at the steering wheel to say vi kører.

If the English sentence simply said "We go through the forest," could I use går?

Only if you are walking! A common trap for English speakers is translating "go" directly to at gå. In Danish, at gå specifically means to walk on foot. By using kører, the sentence makes it clear that we are traveling in a vehicle.