That there are "no go"areas is an exaggeration. "No need" describes it better. That is true for Germany in general. It is not the Bronx in the 70s, nor are the racists and xenophobes patrolling in swarms.
But Berlin: avoid to be in the subway alone, even at day. Incidents have constantly risen there. Avoid antisocials who obviously seek trouble, or drunk or stoned.
Unfriendly, rude tone is part of the Berliners' way of life, at least some of the old Berliners that have not moved in just years ago. But with some others, mostly the younger, it may be according to "rough, but hearty".
The Western city hotspot is around Tauentzien and Kurfürstendamm. It is the major shopping area now. It used to be a paradise for some really great and wonderful movie houses, but they are almost all gone now, a very great loss. The aquarium of the Belrin Zoo. right of it, was great and nice in the 80s, that was the last time I visited it. If you like classical music and a concert is tkaing place, check the Philharmonic, it is one of the concert halls with the best acoustic characteristic wordwide (and has some architecture that many admire, very fast going in and find your seat, very fast leaving, even if it is full hopuse), a bit spaceship like, some say. Second biggest shopping mall is the Schlossstrasse and Rheinstrasse in Schöneberg/Steglitz, but it is just shopping, not culture, and it is more the kind of shopping for the locals, not the big vogue-kind of thing. The Eastern centre of the city, Pariserstrasse and such, is the place to go for restaurants, nightlife and the like - at least so they say, I have little own expoerience with that, it is not my interest so much, and I know the West better than the East. The welathier parts are the former American sector: Charlottenburg, Zehlendorf; the closer you move towards Grunewald and the lakes there, and the Havel in the West, the greener, wealthier, and better looking the city becomes, many villas. The more centre and East of centre you go, the dirtier it becomes. Some people like Kreuzberg and such, I personally hate it. Most overestimated tourist "must sees" are Alexanderplatz and Brandenburgertor, you can safely scratch them from your program, you miss nothing. If weather is nice (unlikely) and you want to do a picnic without leaving the city to the West, Tiergarten has many nice and green corners, its a very huge park. the whole city is surprisingly green and fiulled with small and big parks, you maybe will realise. It's said to be the greeniest city of its size in Europe.
If you move to or through East Germany, consider Spreewald (Google) which is not too far away from Berlin.
When you return from your excursion to Budapest, you maybe move North again, then the oldtown of Wismar and especially Lübeck are the places to see, I think. Very very beautiful old Hanse cities, much restoration done there in the past 20 and 30 years.
General advise for the above: do not rush through Wismar and especially Lübeck. In Lübeck, even take more than just one day. Optionally, you can go from there to Travemünde and the Brodner Ufer at he baltic coast. First is the biggest ferry harbour in the Baltic and has a tourist-attracting pier and holiday city (small one), seocnd is if you would like to walk at the beach, lonely, its is very breautiful a place and you can walk aroudn the whole Lübecker Bay, if you want, all day long, from town to town. If you go to Lübeck, take your time, consider a longer stay. If you go to Lübeck, ask me again, I can talk you through the city, I still know it quite well. Espoecially two or three good restaurants (at least they were good when I ws there the last time. Some atmospheric places also are not that easily to be found. But the old town is a very nice place in general.
US-Americans would consider the old town of Heidelberg to be the place to go in South-West Germany.
They used to crawl over that place like ants.
Just pulling your legs, Yanks!
Sorry, on museums I cannot help much, they simply do not interest me. But there are
many in Berlin, on and close to the socalled Museumsinsel. Any tourism site on Berlin should point them out.
Maybe I can give more advise if I knew what your tourist interests are when you go travelling.
If you are like me and siscover a city by just walking,walking, walking thoruzgh it to get a feel for it, let me know, we can design a kmz-file for Google Earth together that gets you around.
My record day-tour I did on feet in Berlin was in summer 2000, one of the most beautiful and one of these really perfect days. I did a tour of 38 km through seven city districts on that day, in close to 12 hours. Needless to say that I was exhausted. But happy.