Göttingen

As mentioned above, Göttingen's location at the cross of two ICE train lines makes it easy to reach most other major German cities. There are also a number of smaller towns that are worth a visit, which could be seen on a day trip from Göttingen. For the local trains that go to these towns, you can buy cheap group train tickets that can be used for up to 5 people: the €25 Niedersachsen-Ticket is good for all-day travel within Lower Saxony on any day of the week, and the €40 Schönes-Wochenende-Ticket is good for all-day travel anywhere in Germany on a weekend day. You can reach any of these places on local trains in under 2 hours:

Goslar is a UNESCO World Heritage Site, with 1000-year-old silver mines and a palace.

Hamelin of pied piper fame has weekly reenactments of the tale on summer Sundays.

Hannoversch Münden is a small and charming town.

Eisenach is home to the historical Wartburg Castle.

Osterode am Harz is a good starting point for hikes in the Harz mountains.