We went to London and didn't go inside any of the museums or palaces... it was school vacation there, and some of the historical places had severely curtailed hours. We walked for miles, saw a show in the theatre district, shopped, ate, and we did do a one day tour of Stratford-Upon-Avon, which for me, was really work related at the time. DH took a lot of illegal pictures for me at Shakespeare's birthplace (you weren't supposed to use a camera); I used them for the next 4 years with my students.
I went into a few palaces/gardens when I was in Berlin and Pottsdam, but that trip was more for visiting my former exchange student. In Italy, it was all about the eating, scenery, and shopping, though we went into the oldest church in Florence, and one other art museum. We like exploring parts of towns that are off the beaten path. On the other hand, my bike tours in Europe had a lot of museums and history built in... I liked the brewery in the Czech Republic, too! We actually skipped some of the historical stops on that tour, especially on the days it was raining. I just wanted to get to the end of the riding on those days. But, we did do all of the historical side trips planned for after the riding day was over. Last year, when we went to Spain, we did tons of museums and palaces, both before, during, and after our tour.