My wife thinks I should see someone for my rotator cuff syndrome. Should I get x-rayed first?

Maybe. The best people to evaluate this are sports medicine specialists and sports orthopedic surgeons. They may order no imaging, shoulder xray and/or MRI (with or without contrast). This decision is based upon their confidence in clinical exam, symptoms, and age. Older doctors tend to trust their exams more than mri. Xray can be useful to rule out other gross problems or show chronic cuff rupture.
Rotator cuff. The rotator cuff is a group of muscles responsible for shoulder movements. Plain radiographs (x-rays) can't tell whether any of these muscles or their tendons are torn. Studies of choice are MRI or possibly shoulder arthroscopy. See your pcp or an orthopedic or sports medicine specialist for evaluation. (s)he will then order the appropriate test.
YES preferably. To evaluate pain in a region its important you obtain xrays. An orthopedist can easily rule out specific shoulder findings on the xrays such as arthritis in the ac or shoulder joint, presence of spurs and or abnormal bony morphology, subtle fractures, unfused bony fragments etc...Yes xray first then if necessary additional imaging modalities such as mri, mra, ct, cta, bone scan can be ordered.
Yes. Xrays are always recommended to assess the bony structure of your shoulder, and to see if there's any radiographic evidence of what might be responsible for your shoulder pain.