Best train booking options for Prague-Dresden-Berlin-Hamburg ...

Now that all our hotels are booked for this upcoming trip in September, where we will spend 3 days in Prague, one day in Dresden, 3 days in Berlin and 2 days in Hamburg, I am wondering what's the best way to book train travel between these cities? I have looked on the Bahn website but am rather confused if there is a cheaper option than just buying the tickets to these destinations a la carte. Please advise, dear wise Fodorites! :? Posted in the Europe forum.