I have never really heard of cameras outright breaking, so I was wondering if buying a used camera is a good idea. I feel the prices of a new camera is often outrageous, and this would allow me to more quickly get into this wonderful hobby.

Used cameras are fine, you just want to make sure nothing major is wrong with them.

What's your budget?

You talking used digital or used other? If it's digital you can check the shutter actuations ... that will give you some indication of its usage.

If it's not digital then you buy at a little more risk ...you might be able to bring the cam to a repair shop for an inspection.

Either way, I always try before I buy ... never buy used sight unseen. If it's film, run a roll or two, if it's digital, try all different settings and what not. Look for wear and tear such as on the mirror, how dirty the vf is, etc. These all indicate how much usage.

Btw some warranties are non-transferable so if you buy with that in mind think twice, warranties are usually exclusive to the original purchaser.

I thought about buying used lenses, what do I look for? Also, how many pictures can a shutter take?

Lenses always last longer than bodies. Check for scratches and sharpness at different apertures, as well as entry on camera.
Consumer entry level DSLR cameras take anywhere from 2-50,000 exposures. Maybe more.
Buying used lenses is ok. I'm thinking of buying some used lenses and a used rebel to teach a new assistant.