When I pick up the bike so that it is resting on the back wheel and I push on the handle bar, it wobbles as if there is space in the head tube. So I'm guessing there is something wrong with the headset? I wanna find out what kind it has and to see if I can buy parts to have it replaced. Can anyone help?
Depends what you want unsealed or sealed. I can point out two two unsealed that I have liked. The a-head Tank and the FSA Pig headset on the unsealed side. For sealed you can use a Race Face, the new Cane Creek 110 and my favorite the Chris King.
there might not be enough spacers under your stem to keep the headset tight. loosen the pinch bolts on the side of your stem and then toghten down the compression bolt on top, but gently don't over tighten it.
i had the same problem right after i ditched my gyro and brakes altogether. there is too much space between your stem and headset. all you need is some spacers. those will usually run you a few bucks each at your LBS