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You would have probably lifted too much weight that your body could not handle in the first session. We need to first prepare your body for that activity with less load initially in any training. Once your body is ready to take it, you can increase the load by increasing the volume of repetitions. I suggest you kindly ice your elbows now, frequently like 5 to 6 times, to reduce the pain and soreness and follow it with proper stretching. You can take anti-inflammatory tablets after consulting your doctor. Now coming to your climber's elbow, since you are aware of your issue, I suggest you take proactive steps to prevent it by strengthening your forearm and wrist muscles. You can try using a golfer's elbow splint to reduce the load on the tendons of the pronator teres muscle and other wrist flexors.
There are many options as follows.
1) Start icing your forearm regularly, five to six times a day for 15 minutes. If you think you have some swelling, increase icing frequency to once every two hours.
2) Once swelling gets reduced, start working on the forearm and wrist strength. You can use a resistance band to start with and keep loading your forearm with weights progressively after 2 to 3 weeks. Once you think you did good work on strength with good loading for your whole body, start many stability exercises for the shoulder and gluteus muscles. I recommend you to get back to climbing after 6 to 8 weeks. Get a Hydrocortisone injection done under ultrasound guidance by an orthopedic doctor. Work on rehabilitation and strengthening of your forearm and wrist. Rehabilitation exercises that can help you initially are as follows, in sitting position,
1. With the forearm supported and the wrist outside, with the palm facing upwards, move your palm from wrist extension to flexion.
2. With the forearm supported and the wrist outside, move your wrist from flexion to extension with the palm facing the floor.
3. With the forearm supported and the wrist outside the support, the palm facing down do supination. You can wrap the resistance tube around the forearm, and the other end can be held with the opposite hand.
4. With the forearm supported and the wrist outside the support, the palm facing upwards, do pronation. You can wrap the resistance tube opposite the movement performed, and the other end can be tied.
5. You can hold a weight plate, say about two pounds and walk for 20 meters.
6. Stretching your forearm muscle is the key to recovering faster. I recommend you do a good warming up of forearm and wrist muscles in your routine. Do good cool down stretches more for your forearm and wrist muscles.
7. I suggest you check on the exercise form, especially your wrist position while doing gym, as they play a vital role in the muscles under load.
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