I think Budapest is the most underrated city in Europe. I would go to Budapest over Barcelona any day, hands down. I’ve been a couple times now, but these photos are from a day trip that we took from Bratislava. Just a 3.5 hour train ride for only $10 from Bratislava, I consider it a must for anyone in or near Central/Eastern Europe.
The city is split in two by the Danube River and most of what you’ll want to see is along the riverside, keeping your views of the city always wide and open.
The Buda Castle is high priority to walk around, but I would also encourage anyone to pay to go inside. The Fisherman’s Bastion is just a few blocks away, and there are great outdoor restaurants near there and gorgeous views from the hilltop over the city. Next to it is Mathias Church, who’s roof is remarkable. You’ll see the Hungarian Parliament directly across the river, which is worth the short trek to walk around.
My recommendations? Buda Castle, Fishman’s Bastion, Mathias Church, Parliament, any of the marvelous tradition baths, and a meal at one of the city’s ruins bars (I recommend the unique “Racskert Bar”).