If you travel by bus during the week the tickets are very inexpensive. I got to Gothenburg and back for less than $20. It's also only about a two hour drive from Jönköping, so I had no excuse not to visit.
Other than going to the Gothenburg Museum of Art we didn't have anything planned. Their museum has one of the largest collections of Nordic art and also some works by Picasso and Monet. I was more excited to see the latter, but the entire museum was pretty good. It's a bit small, but the admission is free for students so it was worth the visit since we were already in the city.
We spent the rest of our day exploring shops on the main street. We didn't want to get too far from the train station, since finding the museum was tough enough. We got off to a rough start. The GPS' on our phones sent us in two different directions, each claiming to end up at the museum address. We didn't know where to go and decided to rely on our intuition....needless to say it was a discouraging way to start the day. The desire to explore was quickly squashed and we decided not to get too lost after lunch.
Moral of the story: have better directions and research more than one attraction. I will definitely be back to the city though and I'll try to get off the main street.