Two cities across the Danube: hilly Buda with the castle, flat Pest with the boulevards. Thermal baths, ruin bars, and better value than any capital west of it.
Free things to do
Free
Ruin bar circuit, Szimpla first
Szimpla Kert early to see it properly, then follow the noise.
Free
Gellért Hill dawn hike
Up before sunrise, city lights going out under you.
Free
Danube promenade & the Shoes
Sixty pairs of iron shoes on the embankment for the people shot into the river in 1944.