Is it illegal for your mom to keep you away from your dad at the age of 13?

If she has full custody, she has the say. You are approaching the age where you could inquire as to the reasons - which are probably more serious than her not liking him - and ask to visit on a supervised basis. You do not have rights of emancipation yet, so the decision is hers to make. If your dad has custody rights, he could go to court, but it is generally better for everyone to talk and come to an agreement outside of court.