If I am having pain with exercise in my legs, is there a particular doctor I should see about this?

Your doctor. With a few questions, a physical exam, and comparing your ankle and arm blood pressures your primary physician should make a diagnosis. He or she may ask for help from a vascular surgeon, neurologist, spine surgeon, orthopedic surgeon or physical therapist (probably others ) based upon the problem identified.
Yes. You can start with your pcp. He or she will examine you and possibly order some basic tests. Depending on the findings, you may then be referred to a vascular specialist.