ccameselle, you are asking GREAT questions!
Part of the reason that your instructor can hover in a horizontal position with no weights is that he has mastered his posture. I'd be willing to be that he hovers with his back flat, his knees at the same level as his hips, and his head up. Most new divers tend to look down, and have their hips flexed so their knees ride well below their bodies. This combination tends to tilt you into a vertical position. (It also tends to make you bicycle kick.)
When the posture is good, you can play with the position of body parts -- are you better with your arms out in front of you, or crossed under your chest? Is it better to bend your knees, or to extend them?
If you can't maintain a nice, flat position just by adjusting your posture, then you have to move some weight on your rig, but after playing with the body position, you'll be pretty sure where you have to move weight TO. Moving weight UP your body can be done by moving the tank up in the cambands (there's a practical limit to this, because eventually your head will wham into the first stage). Or, if your BC has trim pockets behind your shoulders, you can put some weight there. You can tie a weight around the neck of the tank, or thread a weight onto one of your cambands, or use weight pockets on the cambands (probably not an option for a class, where you are using rental gear).
Do you remember learning to ride a bicycle? At the beginning, you were wobbly at any speed. After a while, you were stable when moving fairly fast, but if you slowed down, you got wobbly again. Eventually, you could bring the bike to a stop and dismount smoothly. It's kind of the same with diving. In the beginning, you're unstable in the water all the time. Eventually, you get to where you're doing pretty well while you're swimming, but if you try to slow down or stop, you lose your balance. It takes some practice to get to where you can hang in the water without moving, but it's a laudable goal, because it really increases the FUN of diving. I remember being very frustrated as a new diver, because my buddy could stop and look at some animal, and I'd have to swim circles around him, because I couldn't do that. It's much nicer now that I can.
So, next time you have a little practice time in the pool, play with your posture. If you're doing your class with a buddy, explain to him what you're going to do, and ask him to sit there and give you some feedback. It's not hard to come up with a hand signal that says, "Flat and horizontal," or "Tilted feet down", or "bent at the hips". A few minutes of playing with it will begin to give you your OWN feedback, as you learn what your body feels like in various positions. Then you can play with moving weight, and see where you balance.
And remember, it's a work in progress. I can be much more still than a new diver, but my cave instructor makes me look like I have no body control at all!