which sites in Budapest would teach me the most about Hungary's history?

Certainly the excellent House of Terror, which covers the period during Soviet rule. Memento Park, a short trip out of the city, has a collection of Soviet-era statuary.

Heroes Square and the Military Museum. Signs pointing to it are in English too. Another site: the St Mathias church.

Museums
The Castel History Museum will take you from Roman occupation through at least the 19th century. I am not a huge museum fan, but I like this one a lot. Its more about describing the birth and growth of the city and less about things: https://varmuzeum.hu/en/
The Hungarian National Museum covers the same period as the Castle Museum but its more about the artifacts. The building itself is history and its somewhat a profound place to visit if you are into Hungarian history. https://mnm.hu/hu
The Holocaust Memorial Center and Museum will show that period: https://hdke.hu/en/from-deprivation-of-rights-to-genocide
The House of Terror Museum will cover Nazi occupation, Hungarian fascism and Russia occupation: https://www.terrorhaza.hu/hu/

A few books to read
Marton, Endre, https://www.encyclopedia.com/arts/educational-magazines/marton-endre-1910-2005
The Forbidden Sky, Little, Brown (Boston, MA), 1971; a first hand account of Russian occupation by a victim of it. Out of print you can usually find a copy on EBay.
Marton, Kati (Kati Ilona Marton), https://en.wikipedia.org/wiki/Kati_Marton
Enemies of the People: My Family's Journey to America, 2009; after her father’s death in 2005 Kati returned to Budapest and using the released communist records put together the story of the life of her parents. In other words a second hand account of her father’s book. Comparing the two is fascinating.
The Great Escape: Nine Jews Who Fled Hitler and Changed the World, 2006; a great revelation of Hungarian brilliance and the impact on the world and the U.S. at the turn of the 20th century.
Sebestyen, Victor https://en.wikipedia.org/wiki/Victor_Sebestyen
Twelve days : the story of the 1956 Hungarian Revolution, 2006; one of the more complete books on the ’56 uprising. Read it and visit the sites discussed in the book.

Places to experience
Every one of the Top Ten Tourist spots in Budapest has historical significance. So its hard to pick just a few. While the city has roots back over 1000 years this is pretty much a 19th/20th century city. It remains intact as it did before WWII and it functions pretty much the same as before WWII. Vestages of the former Austro/Hungarian Culture remain and I guess, for me that is the most interesting. So, on that theme ride the trams, see a performance at the Opera House or the Liszt Academy (do both), visit a grand café like the NY Café, stop at the bar at the Four Seasons Hotel or coffee in the Párisi Udvar Hotel. Walk the length of Andrassy ut, slip into any open courtyard doors, tour the Parliament and the Great Synagogue and shop in a market hall great or small. Wear the city like a nice suit for a week.

Favorite Tourism Video of Budapest https://youtu.be/1nd5AtZIrTk?feature=shared
Anthony Bourdain describing how visually pleasurable Budapest is https://youtu.be/0nd9DuDGCz0?si=aa0c19KaJHEtabkX