If it hurts to grip (make a fist) then it’s actually medial epicondylitis, golfer’s elbow rather than tennis elbow.
My biggest tip would be to relax. I know Josh mentions it in his lessons but being all excited to get started learning bass we might have selective hearing. 
Playing bass (and most stringed instruments) takes very little, like amazingly little, effort and probably most people starting out use way more force than is needed. Try barely touching the strings with your fretting hand and just brushing them with your plucking fingers. It will sound bad or hardly make a sound at all. Slowly increase the pressure of your fretting hand and the force of your plucking hand until you are making good sounding notes. You’ll probably find at that point that you are using a lot less pressure/force than you normally do.
Youtube has a lot of great videos on stretching and exercises for both tennis elbow and golfer’s elbow. Check some out and give them a try.