Can a Jew visit the main attractions in Rome? Including the Vatican, the Vatican museums, the Pantheon, St Peter's Basilica, and the Sistine Chapel. It's a two-part question, first of all, are these considered churches even if they are basically tourist attractions, and second, does it make it a difference that I am going there for the art and nothing else?
You ask regarding a mix of active churches and other Christian sites. You should research which of these sites are used to worship other deities, in which case the halacha below applies, and which sites are of a different type (e.g., museums).
Regarding churches, R Doniel Neustadt answers here that "practically speaking, however, the vast majority of the poskim agree that Christianity is considered avodah zarah and a Jew is forbidden to enter a church."
It is clearly prohibited to enter a house of avodah zarah. The Mishnah prohibits one from even entering a city in which avodah zarah is present. Since it is impossible for us, who live in exile, to adhere to this prohibition, we are considered anusim—under duress—in this regard. Entering an actual house of avodah zarah, however, is clearly prohibited2.
What remains to be clarified, however, is whether or not a church or a mosque is a house of avodah zarah. The poskim are not uniform in their classification of Christians as idol-worshipers. Although the Rambam rules unequivocally that Christians are idol-worshipers, other Rishonim are more tentative. Their view is based on the assumption that non-Jews are considered idol-worshipers only if they totally reject the existence of G-d. Christianity, however, combines the belief in G-d with other idolatrous and alien beliefs. Such a theology is called avodah zarah b’shituf (in combination). Some poskim rule that avodah zarah b’shituf is not considered full-fledged avodah zarah, while others maintain that it is.
Moreover, there is a view that gentiles nowadays cannot be considered idol-worshipers since they are merely following in the tradition of their parents (without actually worshipping idols).