As a student in Aachen you will receive a semester ticket which allows you to use the entire public transport system in the state NRW, but there are many other interesting German cities to visit. Travelling by coach is one of the most affordable ways to connect between cities in Germany. Since the national rail monopoly was abolished in the end of 2012, we have seen a rapidly growing market for long distance bus connections. Usually you need to book your ticket in advance on their website. These companies offer frequent connections between big German cities and are usually a lot cheaper than train tickets:

You can use other websites like http://www.fernbusse.de/ to compare several companies.

If you cannot find a good connection from Aachen, maybe it might help to take a train to Cologne first (for free thanks to your semester ticket) and search for a coach from Cologne, which is much better connected than Aachen.

ADAC PostbusMein Bus