Best way to CURE tennis elbow (tendonitis)?

Get some physical therapy from certified person, if you are not noticing any improvement from self-help.

I'd get one of those rubber shock absorbers for your racket. If you don't have one already of course. But I think tennis elbow is exactly what they're supposed to help prevent.

If you want to keep tennis elbow away, you might need to switch rackets. Heavier rackets which are also head-light (more weight in the handle, less weight in the racket head) are supposed to absorb lots of the shock and be easier on your elbow.

This site has extensive tips on healing your tennis elbow. Stretch your arm a lot and do exercises to strengthen your swinging arm. Follow that link to see some specific exercises.

Once your pain goes away, start playing again, but take it slow.

Resting it. The inflamed tendon fibers stick to each other and pull more. Resting them reduces inflamation and allows the fibers to glide easier once again without pain. Taking Advil will help to it has anti-inflamatory properties.

I agree with Gail, you can't beat a good physio session. Ultra-sound can also be beneficail in repairing damage. It worked well in my case.

Prevention of it happening again: The size of your grip could be causing it. When holding your racket in your playing hand you should be able to fit your index finger between the fingertips and the side of your thumb on your playing hand. Add an overgrip, or get a thinner base grip to correct it if you don't want to get a different racket. I hope that makes sense.