When a person is diagnosed with anemia or trying to get the proper diagnosis- what doctor should they see on a regular basis: the Primary physican or a hematologist?

I would say an Internist can try to treat minor anemia first, and if unresponsive, then a referral to a Hematologist is in order.

Of course, if you feel more comfortable with having a Hematologist oversee your treatment, by all means, ask for a referral.