This is right up my alley. When i was in school i worked for my schools IT department and helped kids liek you out. More than likely your school will not like you hooking up a router, but you should call and ask them. Every school has different policies and different consequences for breaking the rules.
if you plugged your comp in and couldnt get it to work it could be for a couple of reasons. a bad cable, a bad card, a bad port (or they just havent turned them "on" yet, or if your comp was on a network at home you might need to set up for the new network at school. there is an easy to walk through wizard that might do the trick in internet explorer go to tools then select internet options go to the connections tab and then at the top there is a button to setup a new internet connection. from there you may need to run the internet conection wizard and choose an always on broadbad connection and you may also have to run the set up home or small office network.
there are a lot of other possible problems including spyware and viruses affecting your computers connection or the school requiring you to do something special before being able to connect.