Duomo Milan, Vatican, Sistine Chapel & St Peters Basilica Rome Skip The Line Tickets

Can't answer for the duomo in Milan but yes, absolutely purchase advance, timed-entry tickets or tour to the Vatican Museums/Sistine in ROME. You will need to pass through a security check queue but it will move very quickly.

The basilica is free and so the security-check line there can be quite long but moves pretty steadily. You can get around it by booking a guided tour of the museums that allows direct shortcut access to the church from INSIDE the museums via a back door from the Sistine.

http://www.museivaticani.va/content/museivaticani/en.html

Regular, timed-entry entry "open tour" ticket to the museums (this is not a guided tour):
http://www.museivaticani.va/content/museivaticani/en/visita-i-musei/scegli-la-visita/musei-e-collezioni/musei-vaticani-e-cappella-sistina/visita-libera-musei-vaticani-e-cappella-sistina.html

2-hour guided tour of museum highlights. Does not include tour of the basilica but you are allowed access via the shortcut:
http://www.museivaticani.va/content/museivaticani/en/visita-i-musei/scegli-la-visita/musei-e-collezioni/musei-vaticani-e-cappella-sistina/visita-guidata-musei-vaticani-e-cappella-sistina---singoli-.html

3-hour guided tour of both the museum highlights and basilica:
http://www.museivaticani.va/content/museivaticani/en/visita-i-musei/scegli-la-visita/musei-e-collezioni/musei-vaticani-e-basilica-di-san-pietro/visita-guidata-musei-e-basilica-s--pietro-per-singoli-e-gruppi-.html

Editing to add: you don't mention how long you'll be in Rome but the Vatican isn't the only Rome attraction one should purchase advance tickets for. Same for Florence, as it appears you might be spending a day there? Not sure what is on your sightseeing list for both of these cities?