As for Budapest, the major tourist stops should be pretty obvious. Overlooking the Danube river on the Buda side of the city are the palace (and library), the Fisherman's Bastion, and the church of Szent Istvan. Definitely check those out - you'll have to cross the Chain Bridge to get there, and there's a bus that goes that way. The House of Terror (Terror Haza) is on Andrassy Utca (Andrassy Street), which is one of the major streets heading out of the center of town, on the Pest (east) side of Budapest. If I remember correctly, Andrassy Utca feeds into the Square of Heroes, which has a bunch of nice statues, and is in the middle of the embassy district. For shopping, you want to find Vaci Utca in Pest, which is where all the neat shops are. The cool basement wine shop - called Preshaz - is on Vaci Utca. If you can make it down to the south of Pest, there's the Market Hall, which is a huge marketplace with tons of food and produce and such. I had a lot of fun there. Also if I remember correctly, there's good stuff to be found at Oktogon, which is on the yellow line of the subway. It's another "center of town" kinda place, with lots of bustle. Some maps:
From my Hungarian co-worker:
