Can I just buy a new motor and "pop" it in to a chain driven garage door opening mechanism? Or do I have to replace the drive an d pulleys and all that other stuff?

What Model Number? Some you can't.
If care is taken Limits won't need adjusting.

It's a pretty old Craftsman model. I didn't find the model number on it.
Also, it had been squealing very loudly for a long time before the motor gave out. I tried oiling everything I could find, but the noise never changed. Do you think this might be because something was wrong with the track?
I appreciate your help and time.

All Moving parts, rollers, anything that is moved by the Motor should be checked, could be something binding place an extra load on the motor. Did you notice the location of the squeal, any belts?
Does door move smoothly when disengaged from drive.(Pull rope)

The door goes up fine when disengaged, and I didn't notice any squeal at all. Perhaps it was the motor squealing all along. I could never locate the squeal at all.. it was so loud it seemed like it was everywhere :)

We had a problem like this at work we just called B&D roller door people and they replaced the motor no problem but because it was the boss I can't tell you how much it cost but it was quick and effective and the motor has been running for years now without a problem:)