I will be in Leipzig, Germany for four nights. Asking people to give me their thoughts on where I should go for daytrips. Most likely I will go to Dresden for one of the days.
When I was there in 2004, the town of Zwickau (about 90 minutes from Leipzig) had a car museum on the outskirts. Zwickau was where Audi was started, and later, where the infamous Trabant (Trabi) was made. As someone who loves car museums, this was a very worthwhile daytrip. Erfurt is about 2 hours from Leipzig, and I loved it (more than Weimar, which is very close to Erfurt).
As Tim has said, Leipzig is a very interesting city and can keep you busy for several days. Have a look at this earlier thread to get some ideas on sites and activities: http://www.ricksteves.com/graffiti/helpline/index.cfm/rurl/topic/80632/should-i-go-to-bachfest-leipzig.html
Leipzig and outskirts are themselves a rich tourist destination. We found lots to see in Weimar, including a very large garden exhibition (in summer, of course.) There's also the very important concentration camp, accessible by public bus from downtown. I agree that Dresden is very worthwhile, just check the museum closing days. But if you're not into museums, consider not going. We enjoyed Erfurt, don't miss the exceptionally old synagogue and medieval "hoard" exhibited there. Farther afield (you didn't say car or train ... ) is Eisenach, with a (sort-of) "Wagner" castle and lots of Bach sites, if not the actual buildings. We drove a long way (actually from Wiemar, not Leipzig) to Quedlinburg, because I'd always wanted to see the church treasure there (... stolen by American GI's and eventually restored.) It is an exceptionally attractive medieval town in a country with zillions of attractive medieval towns. I wanted to see the Zeiss museum in Jena (cold war optics in most American planetariums, not to mention cameras and bombsights ... ), but we didn't get there. It's certainly a special interest site - my wife majored in Physics, I mean.
I neglected to mention that you should not just visit the Thomaskirche in Leipzig (J. S. Bach's church), but also St. Nicholas Church. It is not a great exaggeration to say that the final toppling of the Soviet Union (in our time) began there, with the brave congregation and the music director of the Gewandhaus, later the music director of the New York Philharmonic, Kurt Masur. I don't mean to be irreverent, but he's old enough that you should read his obituary when it comes up.
If you would consider a 4-hour roundtrip feasible for a daytrip, then look into Quedlinburg. Imagine a bigger version of Rothenburg minus the wall, trinket shops and tour buses, and you have Quedlinburg. I've heard good things about Magdeburg and Halle (2 and 1 hour round trips, respectively), but I haven't visited either yet.
Hi, Leipzig has a greal deal historically and culturally, esp if you're into classical music to keep you occupied. I like the city. If you want to see the monument built a 100 years after the victory of the Allies over Napoleon in 1813, the Völkerschlachtdenkmal, take the S-Bahn from Hbf.
Close to Leipzig: Naumburg - great cathedral with the most famous founders statues of the Middle Ages, nice old town with many Renaissance buildings Torgau - big castle with amazing Renaissance staircase, nice and sleepy old town, first Protestant church of the world, epitaph of Luthers wife Dessau-Wörlitz Garden Realm - first English park on the continent, World Heritage Site (WHS) Wittenberg - home of the Protestant Reformation, Luthers house etc. (WHS) Farther away: Erfurt - best preserved old town of all bigger German cities Weimar - hugely important for German culture, home of Goethe, Schiller, Herder, Nietzsche, the Bauhaus etc., nice preserved old town surrounded by parks and palaces (WHS) Eisenach - the Wartburg is called "the most German castle" and a WHS, nice old town
Harz mountains - many medieval towns like Quedlinburg or Goslar (both WHS), many towns and Germanys northernmost mountains connected by steam trains