Buda
Castle District - Royal Palace, National Library, Matthias Church, Fishermen's Bastion,
Gellért Hill with the statue of bishop Gellért and the Citadella

Pest
Heroes' Square - Millennium Monument, Museum of Fine Arts, Exhibition Hall
Andrássy street, Hungarian State Opera House
Deák square - Jewish Quarter, Great Synagogue, Jewish Museum
St. Stephen's Basilica
Szabadság square - Hungarian Television (former building of Stock Exchange), Hungarian National bank
Kossuth square - Parliament, Museum of Ethnography
Inner City Parish Church
Danube embankment (Korzó)
Váci street,
Vörösmarty square.

City Park - Amusement Park, Budapest Zoo, State Circus and Vajdahunyad Castle
Margaret Island - ruins of a Dominican convent, parks

If someone likes excursion and picnic, it worth to leave the city for a day and have a walk in the Buda hills.