My neck hurts and stays a little stiff, the right side. I had a cortazone shot a few weeks ago. It didn't help. What can I do for it?

Many things. See a physiatrist. They will perform a throrough history and physical examination, order the appropriate tests and then prescribe a rehabilitation program. This may include, medication, physical therapy, particular types of injections (there are several types depending on where the pain is coming from), alternative therapies if needed like acupuncture, chiropractic etc. There are many options.
Pain specialist. You may want to see a specialist who deals with chronic spine pain from a non-surgical perspective. This doctor would be best suited to make recommdations for both conservative (therapy, etc) or different injection therapy and to help make an accurate diagnosis.
Stretches. Sounds like you have some muscle tightness/spasm. If cleared with your physician, do stretches. If it hurts in the back, gently stretch forward. If it hurts on the side, stretch head to opposite side (side bend). Physical therapy can also be effective.