It is not very hard to get around in this rather small town, because most sights, bars, nightclubs, etc. are in walking distance and can be found around the city center (market place).
Another option is to ride a bike, what most students in Goettingen do.
The bus system is wide spread, so this is another option for travelers to reach the outskirts of the town.
And finally there are taxis. Look out for a "PUK-Minicar" (you will have to stop one on the side of the street, because they are not allowed to have a taxi-stand or anything similar), since they have cheaper fares than the regular "white" cabs.