Is perturbation theory usually taught in undergraduate physics, and how much of it is taught in quantum mechanics courses?
Also, how much of quantum field theory would be taught in undergraduate physics?
It really depends on the university you are attending. In the physics undergraduate course at Oxford University we were taught PT in quite a lot of detail in the second year - much more than just the H atom - and went on to apply it in nearly every part of the course thereafter. We were offered optional courses in QFT and other theoretical physics in the (optional) bolt-on Masters year.
However, my understanding is that this is unusual; I know people from other universities that didn't learn any perturbation theory until their third year, and not in the detail that we studied it.
Yes, pertubation theory would be taught in most undergraduate physics courses. As Ron points out this will typically be at the level of the H atom or simple many-body systems. Maybe a bit futher depending on any optional courses.
QFT is typically a graduate level course, occasionally elements of it are options in the final year although not a full treatment. If you are looking at courses and this is something that interests you then it's probably something to ask about