The pain in golfer's elbow is secondary to inflammation of the tendons of the forearm where they insert at the elbow. The inflammation is caused by microscopic tears due to overuse. That is, each time we play some microscopic tears occur in the nonelastic tendon tissue. The body usually can keep up with the rate of healing microscopic tears. But poor technique, or an increase in tennis play, can lead to the number of microscopic tears reaching a point where painful inflammation occurs.
When this happens the most important thing is a period of rest.
Rest from tennis that is. Our arms never really rest just from use at the keyboard and in activities of daily living.
If you don't rest, the risk of worse inflammation continues.
The inflammation is a necessary component of healing. It represents release of chemicals [cyytokines] by white blood cells attracted to the areas of microscopic injury. The chemicals stimulate fibroblasts to secrete protein strands that will eventually coalesce to heal these small tears. The strands initially are like a spiders web - very fine and easily broken down by continued tennis play. Over time they cross link and become strong connective tissue healing the small tears.
Right now very gentle range of motion movements with a red Theraband Flexbar can keep abnormal scar tissue from "glueing" together tissues that should slide past each other.
Only after the pain subsides should you consider doing stronger "strengthening" exercises with a green flexbar.
Return to tennis should be gradual with short sessions that avoid serving. Once you demonstrate you can do longer sessions, it is okay to proceed to serving. Listen to your body. Return of pain means inflammation is again present, and a period of rest then restarting the process over can be done.
Go back and read the numerous posts here on golfer's and tennis elbow. Too often players try to "play through the pain" only to lose many months, even years, when the inflammation takes on a life of its own, and lots of scarring of tendons and muscles occur. Try to avoid this bad experience.
Ultimately strengthening the foream muscles with light dumbell wrist curls, revers curls and pronation/supination exercises can help prevent a recurrence.
Many find there is a less chance of recurrence if they come back with a more flexible frame and soft (natural gut or multifilament) strings. Definitely avoid poly strings, although eventually you may be able to use a hybrid set up after several months.
Take this as a chance to work on your sprint work and agility drills. Everyone needs to do this, and it will let you come back much easier to tennis.
[Of course if the pains are not getting better, get it checked out by a sports medicine specialist.]
Good luck!