Budapest was three towns until 1873: Buda and Óbuda on the hills of the west bank, flat Pest on the east, and the Danube between them, so wide here that the city got its first permanent bridge only in 1849. Since then it has been the second capital of an empire, the scene of a siege that left much of it in ruins in 1944-45, the city that rose against the Soviets in 1956, and the party town of Central Europe, roughly in that order. You can still see bullet holes on the walls in the 7th and 8th districts.
It also sits on hot water. More than a hundred thermal springs feed the city's baths, some under Ottoman domes from the 1500s, and a long soak is as much a part of Budapest as the river. Add grand coffee houses from the 1880s and 90s, ruin bars in half-collapsed courtyards and opera and classical music of a high standard at prices well below Vienna's, and five days fill up easily.
The plan keeps each day on one side of the river. Day one is Castle Hill and Gellért Hill in Buda, day two the Parliament, the Basilica and the old Jewish Quarter, day three Andrássy Avenue, City Park and Széchenyi Bath, day four a trip up the Danube Bend to Szentendre and Visegrád, and day five the Great Market Hall, 1848 and 1956, and Margaret Island. Budapest is also the easiest way into Hungary, and two more guides go further: Eger for castles and red wine and Pécs for the Ottoman south.