You could go to Planned Parenthood or a private practice. I’ve been to both, but now I stick to the private office. My second time at planned parenthood the doctor was NOT professional and I decided I wanted to know who I was going to see every time I went in, rather than getting paired with a random doctor I didn’t know.
If you do decide to go to a private practice I would talk to who ever it is you get your insurance from. It is super expensive without insurance. At 20 you should have already been having paps for two years regardless of sexual activity so you should be to explain that this is a basic health care activity that every woman is supposed to do regularly. As someone said earlier it might be a good idea to say you have painful cramps to your guardian as this often treated by getting on the pill and gives you a prudish excuse to get birth control if you need to or want to use it.
If you decide you want to get the Gardisil series you might point out that it is literally a cure for cancer, something that is usually regarded as a medical holy grail (except when involving cancer in you girlish parts, f*cking double standard).
There is something seriously wrong with the world when getting basic health care (or any health care for that matter) is subject to social criticism. This is a health issue and has absolutely nothing to do with your character or morality, which is none or their business anyways.
This seems like the golden exception to a parents often endless drive to take care or their childrens’ well being. They’ll do anything for health and safety until it gets down to their kids junk. Hypocrites. It makes me sick. </soapbox>