I won't be anywhere near, but let me be your travel guide here for a second.
Berlin: It's a big city and has lots of sights to see, so you have to figure out what you want to see for yourself. But here's a little hiking route that will take about an hour on foot for the short version and two for the longer one and show you some of the different sides of Berlin:
You start out on the east side, on Alexanderplatz which has some distinctly east-german architecture and make your way down Karl-Liebknecht-Str. This will take you across the Museumsinsel (museum island [like old and ancient history? this is where you want to be.]) and right alongside the Berliner Dom. You continue to follow the street (from here on it's Unter den Linden [some nice restaurants, very touristy]) and walk through the Berlin Gate and over to the right, to the Reichstag and the Chancellery.
Short route:
You head left and cross the street, taking you into the tiergarten (Berlin's Central Park) down Ytzhak-Rabin-Str. until you're at the Potsdamer Place (which city planners were sporting to be the new city centre in the early 2000s. All new and has some nice, though a bit expensive bars and shopping.)
http://goo.gl/maps/yvrtC
Long route:
You head alongside the right edge of Tiergarten until you get to Schloß Bellvue (the residence of the German President, which is our guy for the ceremonial stuff, basically the German queen.) You make a left turn and cross the great star, a giant roundabout around the Siegessäule (Victory column, or Goldelse to Berliners).
Take your third right (Fasanierallee). Cross the Landwehrkanal and take a left around the zoo until you're at the Breitscheidplatz, where you'll find the bombed-out memorial church (for locals the hollow tooth) and the two weird new curches they built there sometime in the 60s (the lipstick and the powder box, respectively).
You head back the way you came and cross the channel again when you see it. Then, go right, alongside the channel and through the new embassy quarter until you hit the potsdamer str. Take a left here to Potsdamer Platz.
http://goo.gl/maps/Oh6DK
It's been a while since i've been there, but I remember Kreuzberg being great for pub carwling. I was in a bar there once with some friends and asked the waitress when their happy hour begins. She took a look at her watch and said
hmmmmm, now? So we asked her when it ends and she went
how long do you want to stay?
It was a good evening to say the least. But I hear the area has been gentrified a lot, so maybe someone who's been there more recently can give you a few tips.
Dresden: I always tell people, Dresden is a great place to visit and a crap place to live. I won't go into why it's crap to live there but rather tell you about the
nice to visit part.
What you want to see there is rather condensed. Basically there is no need to head outside this area:
http://goo.gl/maps/Cck3H
This is mainly due to the city being more or less razed by fire bombs at the end of ww2 and only the historical centre being properly restaurated afterwards. But in this Centre there is quite a lot to see (I'll try to list the most interesting things, as I remember them). It's basically a Baroque soveraine's wet dream:
-Semperoper: A fantastic opera house and what you'll see in all the Radeberger ads when you're in the brewery
-Zwinger: The lord's of Saxony-Coburg's residence in the city in the 18th and 19th century
-Fürstenzug: the world's largest tile fresco, depicting all the lords and rulers of Saxony from the 12th century onwards.
-Frauenkirche: Recently rebuilt Baroque church. Still has the bright, pinkish colour of the Elbe-sandstone that is the classic building material in the historic centre. It turns dark when exposed to oxygen, which is the colour of most of the rest of the centre.
-Grünes Gewölbe: Museum that holds the treasures of Saxony's lords. Has some of the most intricate gold work you'll ever see.
There are a lot of other great buildings around there, so I'd suggest you just stroll around and make your own way. But as I said, outside of that 'perimeter' there isn't a lot to see.
Leipzig: No idea, never been there, at least not outside the trainstation.
Potsdam: Well, there's basically just Sanssoucie there. Visit Sansscoucie. It's nice.
Have a nice trip. Let's hope the weather holds.
. Hint: you're probably wrong 
)
