Make sure you visit both sides- Buda (the flat side) and Pest (the hills on th eother side of the Danube). Two ways to get up the hill: hike up by crossing the Elizabeth (Erszebet) Bridge or take the tram, across the Chain Bridge.

Just wandering around will be fun, especially in the Castle area (up on the hill) but one quirky thing my husband and I enjoyed was the Semmelweis Museum, which is a collection in the birthplace of Ignaz Semmelweis, who discovered that the reason women got "childbed fever" and died after giving birth was that doctors weren't washing their hands. More at our site (non-commercial, no pop-ups) at http://www.craftpeople.com/trip.html . Those pictures are form our 2001 trip; we're working on a section from our return in October. We liked it so much we want to go back again!