To Dresden is a short trip from Berlin. Stay at the Hilton or Suitess (slh group) for effortable rates around 100 Euros per night if you book early. Both are located directly at rebuilt Frauenkirche in the historical center, surrounded by an ensemble of wonderful baroque buildings like Semperoper etc. There are many good and not too expensive restaurants around Frauenkirche where you can sit outside.

Leipzig is not really a place I have memories of except the train station. Nothing special as city.

As as alternative to Hamburg I strongly suggest Munich. With ICE high speed train you reach Munich in around 6 hours from Berlin. I.e. dept. 04:27am Berlin Hbf (=main station), arrival 10:45am Munich Hbf (nonstop connection). For many of us Germans Munich is our most beautiful city and with "Southern" mentality. Highest quality of living, very clean and safe. But also expensive.

Look for Deutsche Bahn apps if you have an iPhone or Android device. I book all my train tickets and seat reservations through it. You just need to open a free account, have confirmed it by mail. You get your online ticket by mail to print it out, or what I do, on your phone in your app with a matrix barcode. You present your phone to the DB guy who controls the ticket (he will scan the matrix code) and e.g. your credit card as identity proof (you define this in your account before). The apps tell you also your wagon, seat, track, if delays appear and so on.