My husband and I visited Budapest in September of last year. We have travelled Europe extensively, and I have to say that Budapest is probably my second favorite city (second only to Paris!) It is beautiful, the people are friendly, and it is relatively cheap. . .Mainly, we enjoyed visiting Buda Palace, Fisherman's Bastion, the Parliament Building, St. Stephen's Basillica, the Central Market Hall, the Ethnographic Museum, and Statue Park. We didn't have the time to go, but the Gellart Baths are also supposed to be amazing. We plan on visiting them on our next visit.
We stayed at the Four Seasons Gresham Palace, which was absolutely amazing (and very economical because we went off-season.) If this is a possibility for you, I would HIGHLY recommend staying there. It is probably the best hotel we have ever seen, let alone stayed in. . . .
Good luck, and enjoy Budapest!