Aha! Nice question!
So, let me give you some suggestions. I used to live in Göttingen, Niedersachsen. That was about an hour and half by road south of Hanover. It was well connected to major cities because of its central location. Thus, it was quite easy for me to travel around.
Did I hear you say affordable expenses? Voilà! Here is your friend:
Flixbus – Meinfernbus
You will have amazing offers and it is possibly the cheapest mode of intercity transport, if you are going from one Bundesland to another.
Usually, if you are studying at a University, you will get a Semesterticket that will grant you free rail travel within that Bundesland. For example, university students in Göttingen have free rail travel within Lower Saxony and to Bremen & Hamburg. Similarly, for people studying in Frankfurt, rail travel within Hessen is free.
So, which places to travel to?
The cities such as Berlin, Hamburg, Bremen, and München are must visit. I liked Berlin a lot! But apart from these cities, there are many other places worth visiting. Here are a few:
► Rothernburg ob der Tauber
There are many more places but this should give you a start!