The Kreuzberg neighborhood is a popular section of East Central Berlin roughly delineated by Gorlitzer Park in the east and the Landwehrkanal, a lovely bit of river and greenery that will indulge your desire to take a seat and relax in the midst of all of the sightseeing. And eating. And beer. How neglectful to even think of overlooking the beer.
Some of the best cheap eats in Berlin can be found in Kreuzberg, whether you're exploring international foods re-appropriated for German tastes (think the full range of Asian foods; from incredible falafel joints to Vietnamese restaurants) or trying to wiggle your way into a packed cafe (Morgenland on Skalitzer Strasse, Five Elephant on Reichenberger Strasse, Ankerklause on the river, and KaffeeBar on Graefestrasse are all good examples of such), there are plenty of quality choices for dining. Try your way along Oranienstrasse for a look at a variety of establishments, both dining and nightlife.
As for nightlife, there are a handful of lounges and clubs and a number of bars; many of the cafes also stay open late and serve drinks. Das Edelweiss (Görlitzer Strasse) is a great club for music-lovers, while tiny Farbfernseher (Skalitzer Strasse) has a line out the door all night. Reichenberger Strasse and Skalitzer Strasse also have a good selection of nicely ambient bars and lounges, making Kreuzberg a stellar stop for evening and beyond.