"Maybe four or five cities. Someone told me I should try to go to another country of two, that there is too much of the same in the various German cities."
There is some truth in that. In 1945 nearly all large German cities were rubble, and soon after, nearly all were rebuilt. So it's wise not to focus only on the major cities but also to spend time in smaller ones that were less devastated by the war.
You can go to Belgium or Austria if you like, but no way do you need to.
Do you need a rail pass? For 10 days, maybe, maybe not. Don't worry about that yet. You can buy one when you arrive in Germany if you like. But just for fun, let's say you come up with an insane plan to visit Berlin, Hamburg, Cologne, Frankfurt and Munich in 10 days on a traditional city-to-city-to-city itinerary. That would mean 4 long-distance trips to 5 cities scattered around the country, and a 4-day rail pass might be very convenient and save you some money too (fly into Berlin and out of Munich, or vice-versa, in this case.) It would mean a lot of time spent on the trains too - see map.
But that's probably not the best way to "see Germany." Most experienced travelers recommend 4 days for Berlin and maybe 2-3 in these other cities (all of which were major WW II bombing targets btw) as minimum stays. There's not enough time to dig into these place unless you shortchange yourself.
I think you've gotten the soundest advice so far from Lubitsch, who said this...
Europe is more tightly packed with points of interest and historical
places, there's therefore no need to cross vast spaces. You don't gain
anything by that but you lose time in transit. Don't do more than
three places and use towns as base towns from which you do day trips.
One very excellent base town is NUREMBERG (heavily bombed but rebuilt with the past in mind.) It's a fine city and a major train hub to other destinations. I would recommend 5-6 nights there. You'd have terrific day trip options galore, among them...
Bamberg
Würzburg
Regensburg
Bayreuth
Munich
Bad Windsheim
Iphofen
Day trips from Nuremberg will cost you only €19 - 25 per day using regional day passes like the Bavaria Ticket that you can buy from a ticket machine. You can get to Nuremberg easily by train from either FRA or MUC.
A second base town could be MAINZ for 3-5 days, just a few miles from FRA, and another excellent travel base with great daytrip options...
Middle Rhine Valley, including Marksburg Castle and St. Goar, Rheinfels Castle. Stop in Rüdesheim for a short visit too.
Speyer
Frankfurt
You will find cheap tickets and day passes for these trips from Mainz as well.