Here's another vote for seeing both sides of the city, the Buda (actually this is the side that's on the hill), and the Pest (flat side). Castle hill on the Buda side offers magnificent views from Fisherman's Bastion (free). You can take the funicular up the hill. Take a stroll down the Vaci Utca in Pest, many interesting shops and things to see. Definitely go to a spa; there are many to choose from, they are quite inexpensive, and one visit will make the whole trip worthwhile. I recommend the Gellert Spa located in the Gellert Hotel on the Buda side. Try some goulash, some paprikas dishes, and definitely try some hungarian wine; the Tokai and the "Bulls Blood" are quite tasty.

Have a great trip.