Suggestions for Berlin

Many guides available describing Berlin’s many options. What are your interests?

Look at Potsdam for a day away. Many options there and very close with easy access.

Berlin is quite large with many opportunities for sightseeing. As said above, check some guidebooks first for the main sights, and keep in mind that the major sights are spread all over the city. I stayed in the Hackescher Markt area, which I really enjoyed. It was close to many sights and also close to public transportation so it was very easy to get around.

As ORD Traveler noted above, Potsdam is a good choice for a day trip. It's right next to Berlin and easy to get to. But if you do want to go farther afield, Leipzig is less than a 2 hour train ride away. Since you have 9 days, there is probably time to do both if you want. Dresden is another option although that's a bit further.

If you want to visit the Reichstag building you will need to make reservations in advance.

I just spent three nights in Leipzig and found it to be highly underrated. You would want to travel from Berlin by ICE train; Leipzig is just over an hour from Berlin. Depending on your interests there are opportunities to explore history (Prussian, WW I and II, years in the GDR, the Peaceful Revolution...), art, architecture. It was very easy to get around on foot and with public transit. As already mentioned, Dresden is another excellent choice. It is "prettier" than Leipzig and has top-notch art museums. Both are great choices for music. You can't go wrong with either city!

I'd also suggest the Hackescher Markt area. Walking distance to quite a bit and easy aceess to other areas via public transport.

Berlin is very easy to get around between walking, tram, bus and subway. Revisited Berlin in June after 53 years! What a change!

My favorites:
Bundestag tour https://www.bundestag.de/en/visittheBundestag/dome/registration-245686
Topography of Terror, which is a few blocks from the tourist photo op Check Point Charlie
https://www.topographie.de/en/
Memorial to the Murdered Jews
https://www.berlin.de/en/attractions-and-sights/3560249-3104052-holocaust-memorial.en.html
Book Burning Memorial
https://www.visitberlin.de/en/book-burning-memorial-bebelplatz
Kaiser Wilhelm Church, a memory from that long ago visit
https://www.berlin.de/en/attractions-and-sights/3561433-3104052-kaiser-wilhelm-memorial-church.en.html

Top 10 list of VisitBerlin.

Other city: Potsdam for palaces, Dresden for culture, Hamburg for maritime ambiente (most different to Berlin).

Look at this post as I received many great suggestions for Berlin. https://community.ricksteves.com/travel-forum/germany/what-was-the-best-thing-you-did-in-berlin

I highly recommend a Reichstag guided tour, but you will need to make reservations in advance. Other things I recommend to see in Berlin:

1. Topography of Terror
2. Berlin Wall Memorial (outside so bundle up in January!)
3. East Side Gallery
4. Memorial to the Murdered Jews of Europe
5. Temporary exhibit at the German History Museum (when we were there in September the main museum was closed and I think it still will be in January). The temp exhibit, Roads Not Taken, is really good.
6. Charlottenberg Palace

I think Potsdam is a good suggestion for a day trip. Dresden is quite lovely, but I would want to overnight there.

My recommendation would be to at least go on one good walking tour, so you have an orientation to the city and know where the major landmarks are located. Then go spend more time at them.
2 locations that do not get mentioned very often here are Track 17 and the Olympic Stadium. Not sure if those are ok to visit in Jan. Wannsee Villa might be good.
Potsdam is not too far away and well worth a day or two. It is a charming city. You might even head down to Dresden as it is not too far either.