You will need public transportation in Berlin as major sights are far apart. So it is better for you just to bit the one-week pass (AB zones will suffice, ABC if you want to visit Schafenhausen and/or Postdam) and be done with it. It is a convenient pass, allows use of subways, regional trains, suburban trains, trams, streetcars and buses. Like London, areas of interest in Berlin (and there are many of them) are not tightly packed one next to the other but scattered around.
Beware of some gimmicks about museum passes. I recommend the Museum Pass Berlin (http://www.visitberlin.de/en/article/museum-pass-berlin) AND a separate transport card instead of the bundles, because the bundles (Berlin Welcome CArd) give you a worse value (many discounts you will likely not use).
The Museum Pass Berlin is valid for 3 days and costs 24 Euro.
Now, with all this considered, I'd suggest Prenzlauer Berg as a good residential area that has nice restaurants, bistros and cafes withing walking distance, and a pleasant environment for just walking around in between your sightseeing. Friedschain is nice, but you need to pick up wisely (there is Google Street View to help with that). Mitte is more expensive, somehow closer to major sites but it can be noisy. Moabit has some nice holiday rentals but requires more public transportation to/from your sightseeing. Neuköln stills needs to gentrify more before it becomes attractive and Kreuzberg is too alternative, at least for my taste. Charlottenburg, Pankow, Grunewald and everything west of those areas are too far (makes your trips to major sites look more like a commute than a short hop).
Regardless of area you pick, make sure to pick one with a MetroTram, S-Bahn or U-Bahn station nearby. Check on Google Maps, zoom in and take a look at the closest rail service before commiting to an apartment.