My (totally biased) suggestion:
I would start with 3 days in Berlin to do some sightseeing. Enjoy the
Museumsinsel (Pergamonmuseum, Nationalgalerie, Bode-Museum, etc.) and maybe the "Dokumentationszentrum Berliner Mauer", an exhibition about the Berlin wall. See the Reichstag, the Brandenburger Tor and all the other "must-see"-Spots.
Then I would suggest to go to
Cologne by train (take the direct ICE, 4:20 hours) and stay in Cologne for 3-4 days. Cologne itself has the "
Kölner Dom" (cathedral), a couple of art museums (Museum Ludwig,
Wallraf-Richartz Museum). Do see the "
Römisch-Germanisches Museum", an history museum about the romano-germanic history of Cologne. And much more.
Some ideas for trips from Cologne include
Bonn (capital from 1949 - 1990) for the history of the recent period (i.e. the "
Haus der Geschichte" - the House of the History or the
Beethoven House) or a trip to
Ruhrgebiet to learn about the mining history there and how it has changed the area (i.e. "
Zeche Zollverein" in Essen or the "
Deutsches Bergbau-Museum", the mining museum in Bochum).
In Cologne enjoy the local beer house culture with a "
Kölsch", the only language you can drink as well. Or more. Or an "Alt"
if you dare to order ... ;-)
If you need more information about Cologne, feel free to contact me.
After that take an direct IC train to
Heidelberg (2:40 hours). Heidelberg is a nice town with a small but good preserved middle age old town. And as a bonus: The IC trains from Cologne to Stuttgart are going through the
UNESCO world heritage middle rhine valley. Another option would be Mainz or Stuttgart. In Heidelberg, stroll though the city (
Castle, Old Town, The Philosopher's Way) or go on a tour along the Neckar area. Stay here for 2 nights.
Then off to
Munich (3 hours by IC train from Heidelberg, change trains in Stuttgart).
In Munich there is also a lot to see:
Frauenkirche,
"Englischer Garten",
Nymphenburg Palace,
Deutsches Museum,
BMW museum, Olympic Park (from 1974).
Another idea might be to do some side trips, for example to
Bamberg (beer, history) or
Nürnberg / Nurmberg (history, beer). You may see my bias.
Or go to
Garmisch-Patenkirchen to get on the top of Germany highest mountain with 2962 meter: The
Zugspitze.
Personally I would stay in Bamberg or Nürnberg for at least one night, because I do like to taste local beers and would not want to go back after a nice evening with beer and food. But that's up to you. The trains from Munich to Bamberg take 1:45 (direct ICE, 2 hours with change of trains). Trains to Nürnberg take 1 hour (direct ICE) and trains to Garmisch-Patenkirchen take 1:30 (regional train).