If certain exercises hurt, it could be the result of anything from: loading too much weight; poor form; or simply biomechanics that aren't suited for certain movements. Consider yourself lucky to have isolated the problem exercises at least. For wrist/elbow pain, it's also common to see muscle imbalances that lead to issues down the chain. For instance, if your wrists aren't strong enough to do X lbs on skullcrushers even though your triceps and chest are, additional stress will be loaded on the elbows. Improving grip strength might help offset this, if that's indeed the true issue.
At the end of the day, you have to listen to your body. That doesn't necessarily mean avoiding the gym entirely--especially if the pain isn't long-lasting, but instead of searching for a supplement or other magic bullets (which as a 21 year old, you probably shouldn't need in the first place), switch out certain exercises or lower the weight and see if that helps. e.g., instead of skullcrushers try rope pull-downs; switch up your planks to other variations; etc.