Places to go as far as touristy stuff goes:
Castle District, then again at night.
Wander around and get lost in the older districts in Pest (6th and 7th district in particular).
Take a Danube cruise (in summer).
Visit the House of Terror.
Climb Gellert Hill/visit the Citadella.
Do a tour of the Parliament.
Eat a massive meat meal (ensure there at least 4 different animals on your plate) for brunch (ask Stryke about this, he loved it).
Visit the bunker hospital/nuclear bomb shelter.
Visit Heroes Square (best at night).
Nightlife:
Go to Szimpla and play cso cso (table soccer) against random people who are there (bar/pub)
Go to Instant (bar/pub)
Go to Corvin Teto (more of a club)
Go to Gödör (bar/pub)
Eat a szendvics on the way home at 4AM (or whenever you go)
Walk around and visit random pubs. The above are the well known ones which attract lots of tourists so beers are expensive ($3 for .5L). The random small ones that don't attract tourists have beers for better prices ($1.50-$2).
Random things worth doing:
Tram loop around the city.
Ride the subway.
Walk around Margaret Island.
Go swimming in the thermal baths if that is your kind of thing.
Important note:
Contact me before going cause if I am there I will play tour guide.
Next.
