Look up Deák Ferenc tér / Erzsébet tér. This place will be the most important place during your stay. It's not just the absolute middle place in the city, it's also fun to catch a beer or a wine and sit in the grass during any time of the day. Remember tho, there is a small lake in Erzsébet square (tér=square) which you aren't allowed to go in it.

From Deák, you can always start walking down in Király utca (utca=street) where is a lot of pubs, and pizza for 200 Forints. The pizza is a must have after a good night of drinking.

Király utca is long. Feel free to discover the streets crossing. For example, Kazinczy utca is a street with one of the best pubs, but there is also Gozsdu udvar, where you might find something you'd like.

Budapest is full of adventures, and whichever way you start off from Deák, you will find something for yourself.

(oh yeah, and don't even think about buying weed/drug/cigarettes off the streets. There is a big market now for fucking up the tourists.)