In September, you can expect temperatures to be in the 70s/80s as it is the end of summer in Italy. It may get cooler in the evening towards the end of your trip and further north such as in Venice. You shouldn't have a problem looking like a tourist in Rome, Venice, Capri, and Florence as there are thousands of tourists every day. You may want to keep a lower profile in Naples as petty crimes are more frequent. Basically, if you don't want to look like a tourist, you should wear casual dressy clothes such as you would wear at a corporate job (no sweatpants, sweatshirts and/or odd colors such as pink and orange). Sneakers are ok, especially if you want to do a lot of walking. You may be able to get some good diving in Capri. There are some really nice caverns. Enjoy your trip!