Should I feel self-conscious wearing cycling shorts to school?
I'm currently a senior in high school, so I understand your concern about style. In my opinion, and that of all my friends, wearing lycra shorts to school looks ridiculous, no offense. I don't know how far your commute is, but when I take my road bike to school, it's only 3 miles, so I just wear regular shorts and bring another t-shirt I can throw on so I'm not all sweaty and throw a pair of sneakers into my bag so I'm not stuck wearing cleats all day. If you have a longer commute, just put a pair of basketball shorts on over your lycra shorts. It looks good, feels fine, and others can't even tell you have lycra shorts on underneath them. Baggy cycling shorts don't look much better than lycra when you're in high school, even just for your commute.

As far as changing in the locker room, that should be fine. I've done the same thing plenty of times. If you're self-conscious about it, you can always use a shower stall or something so you have some privacy.