Göttingen is on the Intercity Express (ICE) North-South route between Hamburg and Berlin in the North and Frankfurt and Munich in the South. Current train times can be found here: German Railway (Deutsche Bahn).
Hannover Airport to Göttingen train station. Hannover is the nearest airport to Göttingen (about 120 km). A train link takes you to Hannover central station in 12 minutes (line S5, every 30 minutes) from where you get to Göttingen (Intercity express/ICE train) in about 35 minutes.
There are various trains to Göttingen every 20 minutes but try to take an “IC” or “ICE” train (IC or ICE before the train number). “RE” trains or trains operated by Metronom take far longer to reach Göttingen.
Hamburg Airport to Göttingen train station. Take S-1 tram (direction “Wedel”) from the S-Bahn station at the airport. Get off at the central station to change to a direct IC or ICE train to Göttingen (Trains run every half hour).