First time in Germany; Itinerary help requested

For Frankfurt look at Motel One on Hanauerlandstrasse, Excelsior Hotel near the train station, Ibis on the Friedensbruche, Holiday Inn Express Hauptbahnhof. Or you could do the hostel, Jugendherberge, located on the Main. The Frankfurt Card is a good deal if you want to visit some museums.

For Berlin and Munich, I also recommend Motel One. Good prices and they have AC which might be good to have in Aug.

I would also second Motel One but not all have AC (in Munich at least).

DJ

Except for the rare ICE-Sprinter, such as the ones that go non-stop between Frankfurt and Berlin twice a day, no Germany trains require reservations. Sometimes a reservation is recommended; the German Rail website will show that when you see the schedules.

If you do want a reservation, they can be purchased at a ticket counter or online for about 4,50€ per seat, and I have never seen a train where all seat were already reserved, so you should be able to get reservations right up to train time.

In Berlin, devote an entire day to the German History Museum if you are planning on seeing it. A couple hours simply won't do.

You have plenty of opportunities to see a concentration camp. Dachau is near Munich and Sachsenhausen is near Berlin. Multiple tour companies offer side trips, including Original Berlin Walks and Sandemann's New Europe tours. Your teenage daughter is old enough to see and understand what went on there not so long ago. You don't necessarily need to see both, but seeing one is highly encouraged. If that is just too much, don't miss the Topography of Terror, on the site of the former Gestapo headquarters. It's free and chilling.

Berlin has GREAT shopping, especially for teenage girls. Prices there are a bit lower than the rest of the country and there are a lot of cute boutiques in Prenzlauer Berg, Kreuzberg, and Friedrichshain she'd like.

I usually AirBnb in Germany, but there are some good inexpensive hotels near the Hauptbahnhof in Munich and on Oranienburger Strasse in Mitte, Berlin. Have a great time!

Unless you are locked in to flying back from Frankfurt, why not spend time there at the beginning of the trip and fly back from Berlin? That would free up an entire day that is currently dedicated to getting from Berlin to Frankfurt. Not sure, but the air prices might be better flying "open jaws".

I would also recommend going to Ravensbruck Concentration Camp while you are in Berlin. This was a camp almost strictly for women and after visiting here twice, all I can tell you is that it just feels different there. All the moms, sisters, daughters and grandmas produced a certain aura that infuses the place. Hope that doesn't sound too esoteric, but I don't know how else to describe it.

Have used Insider Tours on 4 different tours in Berlin and also for Sachsenhausen, which has many original buildings as it was used after the war by the Soviets. For a professional, quality tour use Insiders or Original Walks, rather than the free New Europe Tours.