In my case I brought all of my clothes for the week to work. Shoes, socks, dress shirts, slacks. I left them all hanging in my office. I packed an undershirt, underwear and lunch in a backpack and just headed in. My commute is about 9 miles so I wear bike shorts and I have to wear business casual clothes and look semi nice and kept here.
Since youre biking to school and more than likely don't have a dress code, I would just pack clothes in a backpack or paniers and start stomping pedals. Or if you have a locker at your gym, you could leave a set of clothes there like my example and just hit the gym first. If you don't have a locker, paniers are your best bet. You can probably get a good set up to hold all of your stuff for less than $75. If you bike to school, you will make that money back in less than a month. Check out sites like nashbar, pricepoint, rei, and esp. craigslist!