There were a few restrictions inside museums and churches. Pictured weren't allowed inside St. Peter's, but people did anyways. Although, they did so in a concealed fashion. St. Matthias, that's the church in the wide angle shot, was discouraged as well. You are definitely going to have a great time in Hungary.
While in Budapest I recommend going to the town of Eger, where they have interesting wine tasting tours in the valley of the beautiful women. I also recommend going to Szentendre only twenty minutes from Budapest.
Enjoy your trip!