Hey, I live here. Drop me an e-mai, perhaps I can help show you around.
Well, I respectfully disagree with Sivalensis’ opinion. Of course, if I didn’t I wouldn’t be living here for the last four years. I prefer Budapest miles above Vienna.
What to do?
Well, the Castle District for starters, of course. Vaci utca is boring for me. It’s just the classy shopping district of Budapest, and a total tourist rip-off.
The Thermal Baths. Do not under any circumstances miss the baths. Szechenyi is the one I’d recommend for visitors. Gellert is nice and elegant, but Szechenyi is beautiful, half of it is outdoors and tourist friendly. You will pay about 6 dollars to get in, but if you stay less than 4 hours, you get about a dollar back. Less than 3, two bucks back, less than 2, three dollars back. So it’s not as expensive as it seems. For a more Turkish and historical flair, try the Kiraly baths (you go nekkid there. Days for men and women alternate, so check.) For men, Rudas is the place to go.
Hero’s Square and a walk through the park there is recommended.
To relax and people watch, go to Liszt Ferenc ter. There’s a huge coffeehouse/pub scene there. It’s a bit trendy and touristy, but on hot summer days it’s a great place to hang out. Plenty of Budapest’s “beautiful people” hang about there, as well as us mere commoners. 
The synogogue is one of the most famous in Europe, and beautiful. Check it out.
Tips on taxis: Do not use taxis that don’t have a company affiliation. They will charge you 2-3 times normal rates. Instead, you can trust: City, Buda, 6x6, Taxi 2000, EstTaxi, and a smattering of others. Also, visit http://www.usis.hu for the embassy low-down on what scams to avoid. There are a few. Use common sense and you’ll be fine.
The public transportation system is straightforward and cheap here. If you’re staying here for a few days, invest in a 3-day or weekly pass. It’ll be worth your while.
If you like contemporary photography, the Ludwig Museum near the castle currently has a Helmut Newton exhibition.
Margit Island is nice for an afternoon walk and “sightseeing” i.e. seeing half-naked sunbathers of either sex.
Architecture? It’s mostly fin-de-siecle 'round here. The Nagycsarnok at Szabadsag Hid (Liberty/Freedom Bridge) is one of the architectual highlights. Nyugati Palyaudvar (West Train Station) was designed by Eiffel. Ah…here’s a good walk. Start at the beginning of Andrassy Avenue and walk to Hero’s Square. You’ll pass the historic opera house, the ballet institute and a bunch of cool buildings. Andrassy is the premiere avenue of Budapest. Incidentally, seek out Andras Torok’s Critical Guide to Budapest. You’ll find it sold here, too, of course. It’s by far THE BEST guide to the city, with plenty of foolproof maps and suggested walks depending on how long you plan to stay here. You won’t regret buying it.
This is just off the top of my head. I can go on for pages. Any more specific Qs, just contact me.
Pete