Walk up to the castle area or take the funicular train - definitely walk back down as it is very easy downhill! The old town and castle is a small but very pretty area. On one side of it overlooking the river is the Fisherman's Bastion, a folly but pretty all the same.
walk over the chain bridge (British design) and walk the length of Vaci Utca looking at the shps, ****** and lapdancing clubs, and go to the covered market by elizabeth bridge for a lookround.
If you can get to the national museum on museum street (in hungarian) it is excellent and free. Tram goes from elizabeth bridge and stops near it.
Hang on to your wallet and count your change.
If you want somewhere nice to eat, I strongly recommend "Spoon", its on a boat right next the Sofitel Hotel on the "Pest" side of the city, just next to the bridge. Amazing!
Have a nice time, its a lovely place!
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