Should I see a specialist for loss of balance, or can my regular doctor take care of it?

Start with your PCP. If you've not seen a primary care doctor, this is a good place to start. Loss of balance can be due to medications, infections of the inner ear, central nervous system disorders, vascular problems, or even cardiac conditions, so it would be good to start with your regular doctor.
PCP & or Neurology. If you have open access to a neurologist is ok otherwise your pcp may order some blood tests like b12, thyroid function test, syphilis tests and some imaging of the brain MRI (or ct if MRI is contraindicated).Also an ultrasound of the vertebral arteries and carotids can be ordered. Most lack of balance in elderly pts is due to a degenerative ataxic process and all tests are negative so no good tx.