Should I see the doctor for nausea and ringing in ears after working out?

Tinnitus. If it was a one-time occurrence, i wouldn't worry about it. If it happens every time you exercise, i would see an ENT to evaluate both of these symptoms. Feel better.
Yes. You should consider getting a general ear, nose and throat exam with an audiogram to make sure there is no inner ear dysfunction going on, especially if there is any dizziness associated with those other symptoms.