I would go check out Heroes Square. Pretty neat. If you are into heroes, and stuff. Link:
http://en.wikipedia.org/wiki/Heroes'_Square_(Budapest)
My buddy and I walked from any locale that had a beer sign (Stella, Heineken, etc) in front of it, had one or two, then moved onto next establishment. Good times, and get to see more sights (usually twice as many and blurry, but I digress). I would avoid expat hangouts (i.e where only N Americans hangout). If you wanted to hangout with drunk N Americans, you could have stayed at home and done that.
I do second the opinion on the medieval place (forget the name). The barmaids/wenches alone were worth the price of admission, and we spent a while longer there than we should have.
Keep wallet in front pocket, and passport in hotel safe. Whiskey dolly's love to grab your backside (and wallet, while they are there). Don't get money changed in subway, as only drunk tourists do that, and they rip you off (as I found out, even after receiving warnings about it: hey, I was drunk!!!)
Contrary to the doom and gloom scenarios painted by somebody who knew a guy, who knew a guy that got ripped off :
, as long as you are sensible you can have a good time in Budapest (or anywhere for that matter)
AL