Most river cruises include a lovely city tour, taking you to both sides of the city - the "Buda" side and the "Pest" side. The hilly Buda side is more quiet and mostly residential, but offers fantastic views of the city from the tops of the hills, especially from Fisherman's Bastion overlook. Most luxury hotels and other accommodations are on the Pest side (pronounced "pesht"), as are major landmarks, museums, shopping, restaurants, pubs and cafes.
Matthias Church is over 700 years old and is named after the Hungarian King Matthias. The church has a rich history, including becoming the city's main mosque during the Turkish occupation. The church also houses a museum. Both the church and the museum are free to the public.
Fisherman's Bastion is only about 100 years old. The fish market was nearby, and the bastion was built as a tribute to the fisherman who protected the area. The overlook is very popular, as it provides fantastic views of the city and the Danube. Note that a fee is charged to access the upper level.