I spent a couple months in Berlin back in 2004, so my suggestions might be a little dated, but, then again, may not be...
The consensus is that the most interesting neighborhoods in Berlin are in the former east. In fact, you can pretty much tour Berlin's most to least interesting neighborhoods by proceeding from east to west. If you want to see youthful, arty Berlin at full flower, go to Friedrichshain, particularly around Simon-Dach-Strasse, where there are narrow streets filled with beer-swilling 20somethings sitting at picnic tables. This is probably my favorite area for nightlife. Not far away is Karl-Marx-Allee, a gargantuan Soviet-style boulevard lined with Stalin-era neoclassical buildings.
The other most interesting neighborhood in the former East Berlin is Prenzlauer Berg. On Kastantienallee, you'll find a scene similar to Friedrichshain, but with more militant artist-squatters who have taken over buildings and draped anti-capitalist banners on them. Another part of Prenzlauer Berg, around Wasserturmplatz, is much more gentrified and has the vibe of a brownstone neighborhood in Brooklyn (with more colorful architecture); this is where Germany's young left-wing politicians live with their families.
Closer to the core, there's Alexanderplatz, which is worth seeing before it totally loses its uber-socialist character to the makeover it's been slated to receive for years. There's an interesting scene here with trams crisscrossing the pedestrian square, the very Sputnik-era World Clock, and anarchists occupying little plots of ground and trading abuses with the police.
You can generally divide the rest of the city center into two parts (they're confusingly both called Mitte). One is the area around Hackesher Markt, due west of Alexanderplatz along the S-Bahn; it's sometimes called Scheuenviertel or Spandauer Vorstadt to distinguish it from the "other" Mitte. This is an extremely lively area, with a number of restored courtyard apartment complexes converted into sort of art-moderne-style malls, with cafes and interesting stores. Walk up Oranienburgerstrasse from here and see the magnificently restored main synogogue and (if they haven't turned it into a New Urbanist lifestyle center yet) Tacheles, the city's most famous art squat, which remains in its bombed-out state from WWII and has a sort of anti-American museum on the ground floor.
The other part of Mitte is the real center of Berlin, with the baroque monuments of Unter den Linden, the Gendarmenmarkt (a traditional European pedestrian square), the Brandenburg Gate, the Reichstag, busy Friedrichstrasse (restored after 1989 with lots of PoMo and now one of Berlin's busiest shopping streets), and, further south, the very 90s extravaganza of Postsdamer Platz. None of this is very "off the beaten path," and probably worth only skimming if you're not into attractions or have seen them before. The 60s-era Kulturforum, Berlin's Lincoln Center, is way more interesting architecturally than next door Postdamer Platz, and is probably worth checking out while you're here.
If you follow Friedrichstrasse south, you'll reach Oranienstrasse (not to be confused with Oranienburgerstrasse), the main street of Kreuzberg. This was the most bohemian part of West Berlin, hard up against the wall, and it's still largely a mix of 20something artists and where you'll find the nucleus of Berlin's Turkish community.
The rest of what's worth seeing in the former West lies beyond the Tiergarten. The area around the busy Zoologischer Garten station and the Kaiser-Wilhems-Gedachniskirche was the center of West Berlin before the wall fell, and here there are lots of busy department stores, particularly KaDeWe on Tauntzienstrasse, the biggest department store in continental Europe. Keep going in this direction and you'll reach Schoeneberg, Berlin's tired old former gay neighborhood; it has pretty architecture and some active streets and squares, but isn't really what it once was. It's Berlin's West Village or South End.
In the other direction from the Zoo Station area is Kurfurstendamm, famous before 1989 as the Champs-Elysees of Berlin (a title somewhat reclaimed by Unter den Linden today). On either side are Charlottenburg and Wilmersdorf, wealthy, bourgeois neighborhoods filled with fairly decadent Wilheminian architecture. In other words, this is Berlin's Back Bay.
As you can tell from this exhaustive description, Berlin is huge. It feels as big as a city like Tokyo or London, even though it has about 1/10th the population. There are endless neighborhoods stretching in all directions from the core group I described, and you could spend years exploring, searching for hidden gems of early modernist architecture or varied little immigrant communities. If you want to go really off the beaten path, for example, Neukolln, south of Kreuzberg, is a particularly diverse part of the city known for its 50s kitsch design.
As for day excursions, there are a couple of former towns within the Berlin city limits (Spandau and Kopenick) reminiscent of premodern city planning, but I'd recommend Potsdam, which is just adjacent to the city to the southwest; it has some really interesting architecture in the town itself (including a Dutch neighborhood that looks like it was flown in from Amsterdam) and the spectacular Prussian palace complex. It's basically Germany's Versailles. If you want to get out further, you can reach Dresden, Leipzig, or Rostock on easy day trips; all are interesting cities. I even visited Hamburg without staying over - it's only 2.5 hours away on the ICE.
