I've recently had the idea that I might try cycling to work. (To be honest, it's a bit of a scary idea because it isn't something I would ever have imagined myself doing, but I think all the exercise I've been doing is making me a bit mad

)
Anyway, I wondered if there are other people here who cycle regularly who could give me advice on what kind of cycle to use. I currently have an old mountain bike, and I've been getting out and about on it over Easter weekend, and I've been quite pleased with my results distance/time wise. I'm really attached to the mountain bike (kind of in the same way you can get attached to an old pair of jeans), but I have noticed a few problems.
1) It isn't very fast. The best speed I can achieve on the flat without my legs whirling round like crazy is about 18mph.
2) It hurts my wrists after a while because of the weight they're taking, and getting the weight off my wrists hurts my lower back.
3) My legs feel quite cramped up on the pedals, and I can't raise the seat any higher without it getting dangerous when I have to stop.
I'm guessing problems 1 and 3 are because it's designed for off road use. The good thing about it being an off road bike is that the gears go very low, making even the steep hills around here OK to ride up.
I've had a couple of other problems, like the chain coming off and the pedals locking during gear changes, and that coupled with a fair bit of rust and general wear has made me decide I will get a new bike to ride to work, and save the mountain bike for leisure.
So, what would people recommend?
Another mountain bike would give me the same problems listed above.
A road bike would solve all 3 of them (I rode one when I used to deliver the mail, and I loved it), but I've never seen one with more than about 6 gears, and I really need the low gears for the hills.
A racer would (I guess - I've never ridden one) solve problems 1 and 3, and being light and having lots of gears should make the hills manageable. But I'm worried that it would make problem 2 (the weight on my wrists) worse.
Any suggestions?