The Danube does more than look good. It separates two cities with different shapes: hilly Buda on one bank with its castle and terraces, flat Pest on the other with the Parliament building and most of the nightlife. You will cross it more than once a day, and that is normal.
It also helps to know how recently the city became one city. The metropolitan city was not established politically until 1873, even though people have lived here for 2,000 years, and the two halves kept their own characters after the paperwork was done. A morning in Buda and an evening in Pest still feel like two different trips.
The layers stack up in a way you can walk through: Roman remains, an Austro-Hungarian capital, then a Soviet period whose empty buildings became the bars.
Buda: Hills, Castle, Terraces
Buda Castle is a UNESCO World Heritage site and a former royal residence, and the ground in front of it gives you the whole city in one look. Fisherman's Bastion is the fairy-tale piece, all towers and terraces with a view back across the river. Matthias Church is the other stop on the same morning, and both are exterior visits, which keeps things short. Go early if you want the terraces to yourself.

Pest: Parliament, Squares, Evenings
Pest is the flat, busy half. The Hungarian Parliament building sits here, one of the largest in the world, and the Gothic detail is worth a slow walk along the riverside. Heroes' Square is the other anchor, a landmark built around Hungary's millennium, and it is the square that makes the scale of the city click.
Crossing Between Them
Walking plus public transport is the same combination, and metro, tram, bus and trolleybus cover both banks. Line 1, built in 1896, is a UNESCO World Heritage site and the oldest underground railway in continental Europe, so a ride on it counts as sightseeing. Getting around Budapest this way beats a rental car, since parking downtown is limited and expensive.