The Castle District: Cross the Danube on one of Budapest’s famous bridges and you’ll end up in Buda. Wind up the hill to the Castle District and you will be able to explore Buda Castle, the Fisherman’s Bastion, as well as Matthias Church. These destinations are less than 10 minutes by foot from each other, and they will give you an enviable view of Budapest spread out before you--including an unobstructed view to the Hungarian Parliament Building.
As you wind along the streets of old town Buda, there are plenty of cukrászda (confectioneries) and cafes to take refreshment from (Ruszwurm has stood the test of time). Give your legs a rest and take the historic Castle Hill funicular (tickets cost 1200 HUF/4€ one way). On the other hand--if you’re looking for more exercise, hike up Gellért Hill and see Budapest’s Statue of Liberty up close!
Hősök Tere (Heroes Square)--This majestic pillared arc is the gateway to Városliget (City Park). Depending on the time of year you can find a variety of events hosted in the square, as well as the museums on either side which are open year-round. In the winter, meet in the square to walk over to the Városliget ice skating rink, where you can glide in the shadow of Vajdahunyad Castle. In the summer you can walk or tour around the lake in a paddle boat.