Shifting/braking: There are two levers: one is the brake lever, and there's a second lever behind that. You push the brake lever one way and it moves you into a bigger (I think) cog/chain ring. Push the second lever and it moves you into a smaller cog/chain ring. To brake, you just pull back on the brake levers like you'd expect. And I guess as long as you don't brake while shifting, it's all good (BF's advice to me.) It took me a little while to figure all this out. Mine have a thumb trigger, but it's the same principle.

Gears: I have a triple and 8 cogs. I think they go up to 10 cogs, and the big rings can come as a triple or a double. I like my triple, but whatever. The number of teeth on the cogs are probably going to be different--the road bike's largest cog will probably have fewer teeth, since the mountain bike is probably geared for climbing.

If I'm wrong about the above, which is quite likely, someone can come along and correct me.
(And no, you don't sound stupid. I had to ask my boyfriend repeatedly about how it all worked when I was trying to buy my bike.)