Budapest, pronounced Budapescht, consists of two cities, hilly Buda in the west, and flat Pest in the south. The city structure is similar to Prague.
Day 1
Start your day with a walking tour. While there are many specific ones like focusing on Jewish quarters etc. I chose the generic one which covers the history and the culture of the city. After the tour, spend time checking out the castle before heading down to the Hospital in the rock. It was a hospital built during the cold war to survive a nuclear attack. The guided tour at this place is subpar though.
Check out the Mansfeld Peter Memorial and Zero kilometer Stone while going down from the castle.
And then before crossing over to the Pest side, take a bus to Citadella for a few more views of the city.
Day 2
Start the day with St. Stephen’s Basilica. There is a hand of Stephen, the first king of Hungary, inside. It is illegal to construct a building taller than this in Budapest. The Hungary Parliament building is precisely the same height.
Next, head to Shoes on the Danube Bank, which are placed by Hungarians in the memory of the Holocaust victims.
Later at night, check out Szimpla Kert and Red Ruin, the two famous ruin bars among many others.
Note that Budapest is famous for the thermal baths as well. I felt that they are glorified hot water pools with high prices and decided to skip them.