Let's see...
Towns:
Torgau* - a provincial town with a preserved old town (Altstadt) and a mighty castle on the banks of the Elbe. The castle church is the first Protestant church ever, the tomb of Luthers wife Katharina von Bora is at the town church, and there is a big exhibition about the Protestant Reformation right now
Naumburg** - another town with a preserved old town, more impressive but without a castle. The Gothic cathedral however is a top sight and offers some of the best known statues of the Middle Ages
Erfurt*** - the capital of Thuringia offers one of the best and biggest preserved old towns of the country
Weimar*** - a small preserved town with a castle surrounded by palaces and parks (and sadly a KZ too), and a centre of the German culture. Goethe, Schiller, Herder, Nietzsche, Liszt and Cranach lived and worked here, the Bauhaus was based here, and the whole place is just very pleasant.
Halle** - the big city next to Leipzig. One of Germanys most underrated places, even by Germans, but the old town is actually surprisingly diverse and interesting. Go there if you are interested in architecture or want to see the Nebra Sky Disc.
Wittenberg** - home of the Protestant Reformation. A small town with many preserved palces related to the Reformation, like Luthers, Melanchthons or Cranachs house. Due to the coming celebrations soem of these sights could be closed however.
Dessau** - the old town was destroyed in WW2, but the Bauhaus buildings are all still there.
Castles
There are many castles around Leipzig, but sadly most of them are very difficult to reach by public transport. There is the Neuenburg high above the valley of Saale and Unstrut (vineyards) for example, but although the town of Freyburg is both close to Leipzig and Naumburg it takes 90min to get there. The best idea would be to rent a bike at Naumburg and explore the valley and the castle this way.
And there is Colditz castle of course, but again: it's very difficult to get there by public transport, and it takes more than 90min. Colditz is also just one of many (better) castles in this area, but without a car it's almost imposible to see them.
And the Dessau-Wörlitz Garden Realm is highly recommended!