We just returned on the 14th from a 2 week trip. Were in the same cities you mention plus Naples. Check with your airline to see what they will accept. We'd planned on doing a proctored self-test but heard United Airlines wouldn't accept it. I called them and this was true. However, they would accept it for the return trip. We went to an urgent care near us for our test prior to leaving and got a rapid antigen test. Took BinaxNow self-tests with us for testing for our return. Set up the eMed account and downloaded the Navica app and did a trial run a week before leaving. That way we were familiar with the process. Used our cell phones to take the test in our hotel room while connected to the WiFi. Downloaded the results and emailed them to our hotel, who kindly printed it out for us. I can't recall a pharmacy we passed, while over there, that didn't have a sign saying they performed Covid tests. I think I heard it is around 20 euro to get it done. If you decide to take a self-test with you, be sure to pack it in your carry-on or personal bag. The BinaxNow test says it needs to be stored between certain temperatures, with the low end being about 36 degrees F. Cargo hold would ruin a test kit. I assume other self-test kits would have similar storage requirements.
As for crowds, they weren't bad. Venice was least crowded. Florence was busy but we were there on a Friday and Saturday. So there was a decent line at the Uffizzi and quite a few people inside. But not concerningly packed. In Rome, St. Peter's Square was nearly deserted the morning we were there. Trevi Fountain had a good crowd around it so we just put our masks on. Naples had a lot of people on the streets but the only thing we did indoors was the Archaeological Museum and we were about 5 people back when we arrived to buy tickets. Plenty of room to space out once inside. Capri was nearly deserted. Rome and Naples were starting to put up Christmas decorations.
Consider buying a clear plastic pouch/card holder for your CDC vaccine card. You will be pulling it out frequently.