The famous big outdoor spa (seceny?) is incredible. Cannot be missed. Give yourself a few hours to enjoy it. Sitting outside in huge outdoor hot baths whilst it is freezing outside is amazing. All the indoor spas are touristy and expensive.
The old Jewish ghetto a area to the east of the city centre has lots of of really cool (but not ostentatious) bars and little restaurants all of which are cheap and great. Lots of pork and cabbage. There is a restaurant called M which is very good and quirky.
The area down by the Danube is touristy but pretty stunning. There is a big food hall market which is worth a visit to stock up on paprika but is also very touristy.
Walk up to the spa because it is in the really grand old part of town but otherwise don't be scared of the underground system. Reall cheap and fast.
Anyway tl;dr Budapest is an awesome city. Very jealous.