As most museums are closed mondays (an exception is the Deutsche Guggenheim collection of contemporary art at Unter den Linden, which is free mondays, http://www.deutsche-bank-kunst.com/guggenheim/e/ ), just walk a bit through the city to get a feeling for Berlin.
Berlin is divided into many centers,so it's a bit hard to orientate at first. The main centers are in the districts Charlottenburg (west, main station there is "Zoologischer Garten") and Berlin-Mitte (east, main station there is Friedrichstraße).
They built a new main train station (hauptbahnhof) for the whole city between these two, but there is no real center, as it is kind of artificial.
You have to visit the new government quarter, which is located near the new Hauptbahnhof just behind the river spree. Great modern architecture (in my eyes) and there is also the Reichstag where you can walk up to the glass dome for free (just have to wait as there is a queue normally).
Not far away is the Brandenburg gate, it's also the beginning of one of the main streets in Berlin (Unter den Linden). On the other side there is the big city park, called tiergarten (not that interesting in winter).
Another new central part is "potsdamer platz" (located a bit south of the Brandenburg gate), which was built ten years ago as a new city quarter, modern architecture, lots of bars and restaurants and so on, but it's not cheap there.
Friedrichstrasse is the new shopping street in east Berlin, also some nice and not very nice modern architecture.
Great for shopping and just strolling around is the area north of the station Hackescher Markt! Just go where the masses are going ;)
The nicest streets there are Oranienburger Strasse, Neue Schönhauser Strasse, Alte Schönhauser Strasse, Sophienstrasse and Grosse Hamburger Strasse.
Nikolaiviertel (located east of Unter den Linden, you may notice it because of a church with two spiky towers) is the historical heart of Berlin, but was destroyed in WWII and was built as some kind of fake historical architecture in eastern germany in the 80s. Next to it is the red townhall of Berlin and the TV tower of course.
In the evening go to one of the bars near hackescher Markt or around the station Eberswalder Straße (in the district prenzlauer Berg).
Next three days visit some museums with a special three day ticket which is only 16 euro (or so) for normal people and 7,50 euro for students and scholars - you have unlimited access to over 50 museums with this card. Best museums are located at museums island (next to Unter den Linden and the Berlin cathedral) and near potsdamer Platz (ask for Neue Nationalgalerie and Gemäldegalerie for example) and Kreuzberg (Jewish museum and Berlinische Galerie).
Some other nice areas: Oranienstrasse and Bergmannstrasse in Kreuzberg, Simon-Dach-Str. in Friedrichshain (Bars, Restaurants), Pariser Str. in Wilmersdorf, Savignyplatz in Charlottenburg.
a good city map: http://www.berlin.de/stadtplan/map.asp?size=5x5 where you can enter the name of the desired street on the top.