In case you come to Budapest in summer (May - September) you definitely have to visit as many of the so called "romkocsma" (bars in ruins) as possible. They are bars in abandoned houses and keep changing (some closing down, some new ones opening) but those are the ones that I enjoyed most:
Yard Bars:KuplungIt looks a bit "industrial", but is really cool. good electronic music in a big hall and some tables in the yard.
http://kuplung.net/Király utca 46
(map)
Szimpla gardenSome locals say it got very touristy and commercial lately, but it still is one of the coolest places around with a really big garden and a beautiful 1st floor. there's great decoration, random video screenings and even a trabi transformed into a bench and is indie-ish music.
apparently they also have a design market every second sunday.
Kazinczy u. 14
(map)http://www.szimpla.hu/index_en.htm
MumusRecently opened and also really nicely decorated, but the music was not too exciting.
Dob utca 18
(map)
JelenCool underground bar with jazz and experimental music and a really high ceiling. It's actually located on the back side of the warehouse on the corner to stahly utca.
Blaha Lujza tér 1-2
(map)SzodaAnother cool bar in the Jewish quarter.
Wesselényi u. 18
(map)
ClubsCorvintetoIt's located on top of a warehouse and the best place in town for electronic music. You go up in an elevator where you can even buy shots or walk up the stairs. There's a terrace on top of the building with a cool view over the city. Downstairs there's the small but fancy club.
the Entrance is on somogyi bela utca.
Blaha Luza ter 1-2
(map)
Gödör clubFunny mensa-like club with mostly ethno-music. Sometimes they have amazing Balkan Beat Parties. Big and crowded terrace outside.
Erzsebet ter
(map)http://www.godorklub.hu/
TrafoGreat cultural center where you can visit exhibitons, (experimental) dance theatre
http://www.trafo.hu/Liliom str. 41
(map)PiafA really strange non-stop bar. Upstairs you can listen to a women singing chansons with a really funny accent, downstairs you can dance to music you d never dare to if you were sober. Only recommended if you wanna experience something very different or are already drunk.
25 Nagymezo utca
(map)
Places I still wanna go to
A38 (a place for concerts located on a boat)
http://www.a38.hu/tuzrakter (independent cultural center)
http://tuzrakter.hu/site/en/index.html