Anyone can register for a .org website. I would suggest doing some research on servers if you do want to host a .com or .org. At least at my school, you can't host your site on their server with a .org address. It has to be the
www.youruniversity.edu/org/[insert name here] domain. And our servers here are pretty simple. BUT, it may be a good idea to start at your university, as hosting and buying domains may not fit into your budget yet. (Hell, the school is free, even if you can't do much with it.)
I've seen domains as low as $9 a year and web hosting for as low as $4 a month. It all depends on what you want.