What type of permits do I need to convert a basement into a bedroom and bathroom?

You will need the following: Building permit, electrical permit and a plumbing permit as well as fire.

Most likely mechanical as well

I agree. You would need building, electrical, plumbing and mechanical permits. GC can pull all permits under his/her name. You will have a few rough and finish inspections required and based on your State and local Code, you might need inspections for insulation, drywall, shower pan (if shower and not tub) and then final building inspection.

Good luck with your project!