I just got my braces on today and when I was eating, I kept biting inside my right cheek. I expected them to be rubbed by the braces, and sore from that, but I didn't think I would be accidentally biting them when I chew. It really hurts. Is this a normal thing that happens with braces?

Perfectly normal. There is a thing I learned when I had braces: If it doesn't hurt then something is wrong. I know it's weird but with braces pain=good. The more it hurts, the more your teeth are moving and the faster you will see results. The first few months everytime you go to the dentist, your teeth will hurt for about 3 or 4 days. Then the next months it will only be 2 or 3 and at the final sages you will barely feel any pain since the teeth are almost in place. The pain obviously depends on how straight or crooked your teeth currently are, remember th first months are always the most painful. If your braces srratch your mouth and lips like it happened to me, and you start getting cut all inside your mouth, ask your dentist for this kind of transparent wax that you can put on your braces so they don't scratch your mouth.
If you need any more help feel free to ask I can give you plenty of advice since I used braces for almost 2 years.

Thanks for your reply, spandex. I just don't understand why I'm suddenly chomping on my cheek if nothing has moved yet! It's as if my cheek is resting right on the biting surfaces of my upper and lower teeth. I have to kind of smile to get it out of the way. Ugh. I guess at this point it doesn't matter much because now I'm too sore to eat! LOL

You'll be fine tictac. Did the ortho give you any wax? Try that, it really helps. You'll get used to it, trust me.