I think Berlin is a good choice - a historically interesting city.

And it's easy, fast, and cheap to visit other German cities/towns from there.
Potsdam is about 30 minutes from Berlin by train, Leipzig a couple of hours, Dresden also a couple of hours if I remember right and Hamburg is 3 -3 1/2 hours - all by train.

Hotels in Berlin are cheap for Europe and Germany is generally a cheapish destination.

The weather is not too hot in August and Berlin (and Gernany in general) is not too crowded by tourists.