I had/have tennis elbow that I got when I was 18 and, interestingly enough, actually playing tennis... My serve was wrong...
Anyways, the things that helped me were:
1) time...
2) the compression bandages REALLY helped when I was playing tennis. And they helped me recover afterwards, quicker, because I wasn't reinjuring the arm every time.
It's been several years now, and I still take precautions and wear a compression band when I think I might be doing something that'll hurt it, just in case, but I haven't had any pain for at least 6 or 7 years, now.