Cant advise on what to see/do, as there is so much! What we did find was that the public transport is amazing!!! busses, trams and trains, including the underground were a dream to use and got us everywhere. Also clean! Very impressed with Budapest altogether, though there seemed to be a lot of grafitti. People were polite and pleasant though, including the younger element. Not a lot of English spoken, but we got by. We bought Forints in the UK before going, but got a much better exchange rate at the hotel. Euros accepted in some places.
Hope you enjoy it as much as we did! (in our 50s, BTW)