Perhaps you could do research on these clinics yourself, and propose to your mom which one you think would work out, particularly with the free ones. You said you've talked to your mom about it already, just talk to her about it again, ask her to sit down and explain you would like to go to a doctor and try to figure out what times would be best or if she would provide you a ride some day to check out one of the clinics. Stuff probably along those lines.

It takes a lot of initiative on your part, especially when you're going into something so new... but I'm sure that if you meet your parents half way with everything that has to be done to get you there, they might help you easier. Just an idea