Anne, I have been to Budapest for a couple of weeks 2 years ago. See pics in my webpage:
The city is very beautiful even though lately it changed a lot losing most of its eastern flavour. My recomandations are: don't go to big and fancy restaurants. Try small pubs or canteens (they have a light out the street) where the food quality is the same but the prices are much cheaper. 2 years ago I used to have a meal and a beer for less that 5$!!! I like some places near the Oktogon square and metro station. Try the famous goulasch, and if you are sweet-tooth don't miss the local pastry (franciacremes for example): it's just great!
I can't help much about accomodation unless you want to go to a hostel (I worked in a hostel staff).
You should know that the city is divided in 2 parts by the Danube river.
One side is hilly and residential (Buda); there you can find the Castle district, the Gellert hill with a
great view, lots of spas (the most famous is the Gellert one, by the hill).
The other side is more live (Pest), and you can't miss a walk on the famous Vaci Utca. There is also a
nice island (Margareth Island) where you can have a nice walk and sunbath!
On the Pest side is also the magnificent Parliament House.
I hope this helps. Ciao Flavio in Rome