You should probably take your mom with you to your doctor's appointment. It's okay to talk to the doctor yourself, and then have your mom invited into the room. Your mom can also speak privately to the doctor and express to him/her what her own reasons/fears are.
There are a lot of reasons why parents get nervous about tampons. Some have antiquated (and false) notions that tampons affect a girl's virginity (they don't). Some think that a girl shouldn't know too much about her body or get too comfortable inserting something in her vagina, thinking she will be too interested in sex (studies show that the more knowledge a girl has about her body, the more she is able to protect it).
Sometimes parents just don't want to talk about private body parts with their kids - maybe they didn't get much help from their own parents, maybe they have a sense of shame, maybe they are afraid of their own ignorance about anatomy.
Some think that girls won't be responsible about tampons - and that is a legitimate issue! Tampons left in for too long can be a source of toxic shock syndrome, which is very serious and has caused infection and death. So changing tampons frequently is really important. My worry about girls who have to consider paying for their own products is that they will try to save money by leaving tampons in for 8 hours! Proper disposal is also critical for plumbing protection!
I don't think you should go behind your mother's back. She will find out, because she'll keep buying pads for you and wondering why they aren't used, and you will have to deal with tampon disposal without her finding out. And, it will set you up for her distrusting you all through your teen years. Not a good way to start out when you will be facing issues such as driving, dating, curfews, parties, sleepovers, and so on.
The best thing to do is to see if your doctor will educate her, or to see if there's a trusted woman in your life (an aunt, a friend's mother) who can help talk to your mother. Your doctor, or your doctor's nurse, can even help you practice once or twice to be sure you are inserting the tampon properly (a lot of girls are afraid to put it in far enough because they are afraid it will get "lost" - it won't! There's no place for it to go!). There are also different styles of tampons - different absorbency levels for different times during your period, different applicator styles (and some with no applicators) - so you might need to experiment a little to see what's best for you.
There are also great books - the iconic book to educate women many years ago was "Our Bodies, Ourselves" which has more info than you can possibly imagine. There are many others I'm sure - you could look in the library or at a bookstore, or ask the school nurse for some information.
The more you and your mother are educated, the better it will be for you. And this could be the start of some nice conversations between you if you can try to learn more about her position and to be understanding and mature about it, and if she is willing to learn more, confront/express her fears, and start to see you as a young woman and not a kid.