Gellért Hill is the lump of dolomite that the whole city uses as its compass — you can see the Liberty Statue from almost anywhere, and from up there you can see almost everywhere back. This is the climb, the panorama, and the bath at the bottom as the reward.
Around three and a half kilometres with real height, best in the morning before the heat. Start at Szent Gellért tér by the art-nouveau Gellért fürdő (thermal bath) — note it, don't enter. Just above, the Sziklatemplom (cave church) is cut into the rock, walled shut under communism and reopened in 1989.
Climb past the Szent Gellért Monument, where legend has pagans rolling the bishop off the cliff in a barrel, to the summit. The Citadella — a Habsburg fortress built to threaten the city below — has been under long restoration, but the terrace and the Szabadság-szobor (Liberty Statue) above it give the widest view in Budapest: both banks, every bridge, the lot.
Drop down the wooded eastern garden to the baths, and earn the warm water you walked past two hours ago.