Budapest Beyond Sightseeing

The first thing I would recommend is this site.

http://www.beyondbudapest.hu/english/tour_socioculture.html

They organise trips for tourists too (in English). If you are looking for a sightseeing tour that shows you the real thing and not the usual sights you can find in a guidebook you need this.

There's a famous (rather notorious) district in Budapest: the 8th district. This is the poorest part of the city which used to be rich. There are a lot of bourgeois mansions now in a very bad condition which used to be beautiful.
These trips show you how to see the finery that once was. They guide you into actual people's homes and let you listen to their stories which is by far the best idea I ever heared of. What do you think?
There are different kind of trips in different topics. Check it out and decide.
Unfortunately there are very few dates of the tour but if you can bring some more friends with you (and you are minimum 5 people) you can book a private tour date.