Should I see a specialist for rotator cuff syndrome, or can my regular doctor take care of it?

Specialist. See an orthopedic shoulder specialist, who can pinpoint your problem much more readily than your regular, or family, doctor.
Good Question. Impingement syndrome as a diagnosis is increasingly controversial and i don't use it as a primary diagnosis. You can learn more about this topic here: http://www.Theshouldercenter.Com/shoulderpain/2013/shoulder-surgery/spur-acromioplasty-impingement/.
Start with PCP. Most rotator cuff problems are tendinitis, which is treated conservatively - relative rest, ice, meds for pain and physical therapy. If those treatments fail, consultation with a physiatrist or orthopedic surgeon is often helpful. An MRI is often ordered at that time. Even if the rotator cuff is torn, it might not require surgery.