I would talk to my mother and see if I can get going on tampons. You certainly have a good enough reason. It is also possible for a nurse or nurse-practitioner at your doctor's office to explain how to use them.
If you don't want to use tampons, then you're going to have to tell your teacher that you are on your period and that you won't be participating in class for those couple of days. Maybe a quick note from the school nurse will be good enough. And remember you may not be the only girl with their period.
You don't have to use them if you don't want to.
I'm pretty sure I went through this in high school too. There were tons of girls who sat out due to their period, but yes, you do have to tell your teacher. For me I guess it was easier because it was a female. I didn't start wearing tampons until I was about 28/30yrs old because they were so uncomfortable.
Most of the time when you're in water your period doesn't leak out anyhow...but if its heavy it COULD pour out when you get out of the water.
In my high school we have a swim team. I'm also not a big fan of using tampons, so I tried swimming without them keeping in mind my practice is an hour and a half. After a while if you get out and do walk around it will start to run down your leg just because the water is cleansing you it may over cleanse and start to drip, but i would not recommend using tampons for swimming over half and hour.