Berlin Home Base

Sites in Berlin are very spread out, so there's no neighborhood that's close to most of the sites, but if you had to pick one that is near many sites, it would be Mitte. But it is not the most interesting neighborhood, so if you're planning to use public transportation (which is pretty much essential in Berlin, and also very convenient), you might look for an alternative.

I'd suggest searching Google for "best berlin neighborhoods" or some variant on that, and read about some of the interesting areas of the city.

Centre = Mitte
Hipsters = Friedrichshain/Kreuzberg
Gentrified former Hipsters = Prenzlauer Berg
Old East = Köpenick/Treptow
Old West = Charlottenburg/Schöneberg
Little Baghdad = Wedding/Neukölln
Suburb with lots of money = Wannsee
Suburb with not much money = Köpenick
I want to see Berlin, but don't like big cities = Potsdam

A great base for us was on K'damm within a few minutes walk of Uhlandstraße U-bahn station. We stayed at a Kempinski hotel. This location is convenient to public transport especially the Zoologischer Garten Rail Station, KeDeWa, the Tiergarten, the Käthe-Kollwitz Museum and so much more.

I stayed in the Savignyplatz area of Charlottenburg, convenient to both the Savignyplatz (S bahn station and the Zoologischer Garten (S and U bahn) station for getting around to other areas and to the main train station. Close to K'damm shopping street and many restaurants and comfortable to walk around in the evening. I was a senior female traveling solo and always felt safe when out and about. If you plan to take any of the Original Berlin walking tours most of them meet at Zoologisher Garten so very convenient for that also - I highly recommend their tours. I would definitely stay in that area again.

Wealthy suburb...Charlotttenburg, Zehlendorf,

Old east....Karlshorst, Köpenick,

For something very historic and different in the west....Potsdam and Spandau.

We have twice stayed at family-run Hotel Pension Columbus am Kurfürstendamm in the Charlottenburg area, only blocks to U-bahn, Zoo train station, bus to Museum Island. Felt very safe there.

The RS Germany tour ends in Berlin. We stayed at Myers hotel in the Prenzlauer neighborhood. U bahn station was 2 blocks away and tram station one block away. Alexander Platz was close by and it seemed llike you could connect to anywhere in Berlin from there. Neighborhood felt very safe with a nice selection of restaurants nearby.

Parts of this article may be useful in planning your visit -
What to Do in Berlin, According to 6 Locals

A cast of creative people, including a magazine editor and a chef, share their favorite places to eat, drink, shop and play.

Berlin has long been the world capital of the underground. It’s a city where artists, musicians, writers and misfits can find — and afford — a home and studio space somewhere within its massive 344 square miles (for perspective, that’s about nine times the size of Paris). These days, though, the German capital is known as much for its tech start-ups, cutting-edge restaurants and beautifully designed hotels as for its avant-garde electronic music and gritty art scenes. Here, six locals share their favorite spots.

Please forget all entries naming districts with adding a label. For me as a local and born here especially post #2 and two others are cliché prejudices and painful to read because it is simply not reality. Berlin is administrated in districts but not lived that way. Even administration recognized that and changed the responsibilities.

In and across all districts you will find "Kieze" (neigborhoods) which you may like or not. Very often one or two streets away make a difference and you are in the next Kiez. They also have their own characters and events. Example for a Kiez festival in Mitte: https://www.visitberlin.de/en/event/moabiter-kiezfest-lokal-kreativ-bunt

Mitte is extremely heterogenous (see example above and also Moabit is not one Kiez, I know three there). In your case I suggest to focus on two areas for close to sights and good connected to public transport with reasonably pricing:
1) Between Hackescher Markt and Rosa-Luxemburg-Platz
2) Around Spittelmarkt

In Berlin I always recommend and ask visitors to use hostel or hotel if you want to travel responsibly - in terms of community (avoid dislodging low income families).

Tip for date of travel: If you like multi-cultural events plan your stay around "Karneval der Kulturen" / carnival of cultures. Date 2019 open. https://www.karneval-berlin.de/en/

I stayed in Prenzlauer Berg neighborhood in an apartment on Wichertstrasse. Very close to Prenzlauer Alee S bahn station and the tram. Nice safe neighborhood with plenty of markets and restaurants. Took about 15-20 minutes to get to center on tram. Be sure if you rent an AirBnB that the property is registered properly - I found out mid-stay that my host had not followed the legal requirements (the penalty would have been on her, not me, but still it made me uncomfortable!)

Mitte worked well for us. We stayed near Hakescher Market and walked to many sites.