You are quite lucky :) Halle has been described as the Diva in Grey and is right in the middle of Germany, with loads of things to do around Halle.
There is a non-active volcano outside Halle (scientific info in English here and here), where a monastery now stands on top.
Leipzig is the next biggest town (and the place where Mr. Splendid was delivered). It is famous for its trade fairs (with merchants selling their goods in Bucharest to the Sunshine's great-great...great-grandparents), for J.S. Bach, the Monday demonstrations in ’89, its zoo and parks, Auerbach’s Keller (J.W. Goethe), Napoleon’s defeat…
It is about half an hour away on the motorway by car from Halle, and you can easily spend half a day in Leipzig.
Dresden is farther than Leipzig but equally if not more famous. It was the place of the courts of the Saxonian kings and hence has some magnificent buildings to offer. Among these are the Frauenkirche, the Grune Palais, and the Zwinger. The Zwinger also holds a splendid exhibition of art, including paintings by Michelangelo. The landscape around Dresden, particularly along the Elbe river towards the Czech Republic, is to be recommended for a trip and walk. It is called the Saxonian Swiss because of its many cliffs, caves and magnificent views.
Dresden is about 1h 15min from Halle by car on the motorway and takes about one day (or more) to visit.
Berlin, well, it’s Berlin.
It is circa 1 1/2 hours from Halle by car (on the motorway of course, there are no other roads in Germany…). You can do many things in one day and be back to Halle in the night.
While these three are bigger cities the area around Halle also has some smaller cities and more landscape to offer.
If you like wine and want to try some, you just have to drive half an hour and you will be in the Saale-Unstrut valley, and see cities like Freyburg, Naumburg, or Bad Kosen. The region produces some good wines and also sparkling wines, and it is here where Mr. Splendid spent his summers with his grandparents. Moreover, Nietzsche lived in Naumburg for a while, which has also an early Gothic abbey to offer. Freyburg, the actual place of the summer holidays, has a castle, some woods and the most vineries to offer. Bad Kosen is a spa resort, where salt water is used for the treatment of Asthma or Rheuma; you can get nice Windbeutel (puff doughs) there but will have to share with elderly people on treatment.
If you rather like more culture then Weimar will be the place for you to go. It is here where Goethe and Schiller lived most of their lives and it is today a renovated and rejuvenated city celebrating its two famous offsprings. It is about 1h 15mins away from Halle and you can spend 3/4 of a day there. It is has nice gardens and good museums on Goethe and Schiller. You can also have a walk along the Ilm river.
OK, if you still don’t know what to do then it might best to send you off to hiking in the mountains. The Harz region is about one hour away from Halle and offers the highest mountain in central Germany (more than 1000 meters). Around that mountain, you will find woods and other, smaller, mountains. A day trip is doable and the area is more or less touristically discovered.
And finally, Halle itself is a nice city (you either love it or hate it) and along the river, it can be very nice indeed (as the British would say :) ).
