Dessau sits at a useful junction in central Germany and several worthwhile destinations are within an hour by train.
Lutherstadt Wittenberg is 40 minutes east by regional train — the city where Martin Luther nailed his 95 Theses to the church door in 1517 and where he lived, worked, and is buried. The old town is a UNESCO World Heritage Site and a pleasant half-day: the Castle Church, the Luther House (now a museum), and the market square with its Lucas Cranach connections. Manageable in a single morning.
Halle is about an hour northwest — a larger university city with a lively arts scene, the birthplace of Handel, and a well-preserved medieval market square. More variety than Dessau in terms of restaurants and cafes. Worth a half-day or a full day if you combine it with the Moritzburg art museum.
Leipzig is an hour southwest — one of the most dynamic cities in eastern Germany, with excellent museums (Museum der bildenden Künste, Zeitgeschichtliches Forum), a strong music heritage (Bach, Mendelssohn, the Gewandhaus), and a genuinely good restaurant and bar scene. A full day minimum; easily combines with an overnight if you want more time.
Wörlitz village, just outside the Garden Kingdom park, is not so much a day trip as a useful extension of a Wörlitzer Park visit. A small settlement with a few cafes and restaurants at the park entrance — the right place to stop for lunch before or after a morning in the gardens.
