My husband and I spent a whole afternoon in the Castle Hill District which is located on the Buda side of the Danube River. It offers scenic historic walks. We used a Hop-on Hop-off bus to get there. The Castle District is a part of Budapest's Heritage Site. Castle Hill offers medieval buildings and Barogue cobblestone streets. There are great views of the city, the Danube, and surrounding countryside. This area was reduced to rubble during World War II and is now restored. While we were there we visited Matthias Catholic Church that we found worthwhile.
We also got to watched the changing of the Guards at the offices of the President which was interesting and entertaining.