These ideas were shared by my friend Beth from college who spent a semester in Berlin:
sites: brandenburger tor (brandenberg gate), reichstag (old legislative building, cool “green” addition with a good view), and holocaust memorial (a few years old) are all very close to each other.
the street leading there is unter den linden – very pretty, tree-lined, and very pricey. potsdamer platz is also near-by—the super-commercial attempt to “globalize” berlin, not very interesting. (except for a really nice, modern philharmonic building, where there was recently a fire)
fun neighborhoods: kreuzberg, prenzlauer berg, and neukölln.
my fave street is kastanienallee in prenzlauer berg (where i lived) —check out morgenrot for veggie food (especially weekend brunch if possible!) and radical politics. also Babel, for the best falafel i’ve ever had (lebanese) and “der imbiss”, which has an upside mcdonald’s M as it’s logo and delicious naan pizzas (the name is stolen from those little carts that are all over the city – “the snack” – so asking around for it might not be productive, but the logo stands out. near the corner of Zionskirchstrasse).
the strassenbahn (streetcar) goes down kastanienallee and passes all of these places. you can also turn and walk down schönhauser allee for more shops etc. the mauerpark (wall park), a portion of where the wall was, is very close to there also. there aren’t wall remnants (read: not touristy), but it’s a nice park, sometimes people play music, sundays there’s a flea market.
note, you can always get cheap falafel on the street from vendors, but it’s usually nasty, premade stuff – watch out.
kreuzberg has large turkish and hipster populations. neukölln is the new wave (more turkish & arab, increasingly hipster). kreuzberg is probably the better of the two for wandering around. try oranienstraße—U1 to Kottbusser Tor.
there’s a fairly american bar called 8mm, which i liked ( 177b Schönhauser Allee, via U2 to Senefelderplatz). white trash is another popular bar/club that’s nice.
I think 8 mm is not far from The Circus, the hostel i know people have rated well.
the “fernsehturm,” a ussr-era tv tower at alexanderplatz (tall, ufo-looking, you can see it from almost anywhere in the city) has a terrific view – i’d say it’s worth the 7 or 8 euros.
my favorite history-related museum is the museum of the resistance/Widerstand