How can you relieve pain from tendonitis in the elbows?

Conservative mgmt. Modification of activities, counter force bracing, elbow therapy focused on strengthening and modalities, and anti-inflammatories vs a steroid injection (if indicated and you are a candidate).
RICE. Rest, ice, compression or bracing, and elevation. An anti-inflammatory (oral or injection) may be needed. As well as some physical therapy.
Start with RICE. Tendonitis (inflammed tendon) commonly results from overuse. Start treatment w/ rice, nsaid, stretch/ strength prog (pt or ot), & consider couterforce bracing. Massage may also help. If these treatments fail you may be a candidate for cortisone/ prp injections. Few people end up needing surgery.