I lived there from 1998 - 2003, last visited in 2007, so I could help you out.
First, for food, http://chew.hu is a great English-language resource. The Top 33 list link at the top of the page is very well researched and dependable. I would say, if you want Hungarian food, Café Kör comes highly recommended from me (and it’s been on that Top 33 list since I lived there). It’s also not far from Liszt Ferenc tér (Franz Liszt square), a great spot for taking in the Hungarian coffeehouse culture and people watching. It’s a really bustling area during the summer, and the Franz Liszt musical academy is right there. If you can catch a concert there, you should definitely do so.
Do not miss a visit to the famous thermal baths of Budapest. My favorite tourist-friendly one is the Szechenyi Baths, located in City Park (Varósliget) near Heroes Square (Hősök tere). The place is huge and can be a bit of maze to navigate: not only do you have the big outdoor pools (there’s two thermal pools and a swimming pool), but you also have saunas, steamrooms, massage rooms, and additional thermal baths, etc., on the inside. If you’re a little more adventurous and want something even more historical, you might want to check out the Rudas Baths, a Turkish bathhouse dating back to the 16th century. It used to also be a place to spot Bobby Fischer back in the days he was hanging out in Budapest. Unless things have changed, the baths are single-sex, male only. (Although I believe there is a pool section that allows women.)
Let’s see, if you like transportation, Budapest is home to Continental Europe’s first underground, the Metro 1 (Yellow) line that runs from Vörösmarty tér (just east of the Danube, smack dab in the middle of the touristy/shopping district) up Andrassy út (Budapest’s grandest boulevard) to a terminus just past City Park. It’s quite cute (basically, it looks like an underground tram rather than the big hulking subways of Metro Lines 2 & 3). A reasonable itinerary could be to enjoy a coffee and cake at the Cafe Gerbaud on Vörösmarty tér, take the #1 line up to the baths, enjoy the baths, walk back down through Heroes Square (where you have two art museums nearby to choose from: the main Budapest fine art museum (Szépművészeti Múzeum) as well as a smaller contemporary art museum, the Műcsarnok. I’d peruse an art museum, then continue my way down Andrassy, passing Franz Liszt Square and the Opera house to Café Kör for dinner. Here’s my suggested route. That should take up a good bit of your day.
If you’ve still got energy at this point, keep walking west, and cross the Danube on the Chain Bridge (Lánchíd), Budapest’s most famous bridge. When you cross the bridge, there is very nice pub, the Lánchíd Söröző, located right off the traffic cricle, to your right. If you’re thirsty, stop by for a couple of pints there. Despite the fact that it’s located right by the Chain Bridge at the base of Castle Hill, it’s actually quite populated by locals. This was my local hangout and, in my opinion, one of the nicest bars in Budapest. Be sure to try some of the local hard alcohols (if you’re into that sort of thing). Unicum is famous Hungarian bitter which is probably most comparable to Jagermeister, but less sweet and more bitter and herbal. Also, ask for the házipalinka (“hozzy-paw-leen-kaw”), which is the house brandy, a clear distillate of fruit juices, with fruit varying depending on the producer, but can feature plums, peaches, pears, etc.
If you don’t feel like stopping for a pint (or after you’ve refreshed), you can continue heading west onto the path which will lead you up Castle Hill into the part of Budapest I like to call “Disney Buda.” Or, if you want, you can take the overpriced (IMO) funicular up the hill, just so you can introduce that word into conversation. Once atop Castle Hill, there’s a very good contemporary art museum that I enjoyed, the Ludwig Museum. This is just a good part of Budapest to lose yourself in, but not so much to eat and drink in. Wander around the castle and Fisherman’s Bastion. If you’re lucky, there may be a free organ concert at St. Matthias Church (highly recommended.)
Forgive me for rambling. I can go on for much longer, but this should give you a starting point. If you have any specific questions, I’m happy to help. My knowledge is a little outdated for the more hip hangouts. If you do like pub culture, Budapest has exploded with courtyard beer gardens, which started becoming popular in about 2003, right when I was moving out. Szimpla kert was one of the first, opening up in an abandoned building in an old section of the Jewish Quarter. Oh, actually, looks like there’s a whole website devoted to these types of pubs, called Ruinpubs, apparently.
Anyhow, let any questions fly.