I like historical sites, artistic venues, and majestic scenery. Outside of admiring the large cathedrals in all the major cities:
- Budapest: the opera house if you like that, definitely the Hungarian Parliament building, Castle Hill (great views from there), and believe it or not there's a tiny little park named for Elvis Presley in Budapest. It's nothing to look at, but it's nice to know about. Elvis donated money to support the Hungarians in the 1957 resistance to the Communists. A narrated cruise on the river is nice. (There's also a museum dedicated to Houdini's life -- House of Houdini -- where you might find a plaque on the wall commemorating my pre-opening visit in 2016 when I was president of the largest magicians' society in the world.) If you're a nerd like me, the National Széchényi Library is impressive.
Added:
https://www.travelyesplease.com/travel-blog-budapest-castle-hill/
- Vienna: Schönbrunn Palace. The Hofburg if you have time. Just walking around there in the city center -- you'll see interesting stuff around every corner. It's a nice train ride from Budapest - you can go to Vienna for a few hours, walk around the city center, then come back to Budapest the same day.
- Rome: one of the great cities of the world. Take a Vatican Tour, take a narrated/guided tour of the Colosseum and the Forum, and if you need hotel recommendations I can provide some but it's pre-covid.
- If you have the option, I highly recommend a visit to Venice, even though you didn't mention it above. Tour the main sites, coffee or drinks in St. Mark's Square, and if you have time, visit Murano (glass) and/or Burano (lace).
Tip of the iceberg. So much to absorb in all these cities...