...but we also want to experience some of the real Italy, and real
Italian life.
The good news? It's ALL "real". Any place where Italians live, work, raise families, go to school, etc. is "real". Granted, a few locations, like Venice, are experiencing difficulties with a large number of residences converting to holiday rental units, hotels and B&Bs but for the native population, it's still "real". So, city or small town, observing Italian life is simply a matter of keeping eyes and ears open and wandering off the tourist routes. Food? You'll find it everywhere as well, with different flavors according to region.
Florence is a very good base: it's a major transport hub for a number of day trips further afield and a fascinating city in-and-of itself (the art, the art, the art!!!!). I'll stop short of labeling Pisa a "trap" but will agree that there are more interesting places to explore: Lucca, for instance, is a preferred day trip versus Pisa with RS posters who've done that one: you might enjoy hoofing or biking the old city walls.
https://www.visittuscany.com/en/ideas/the-green-city-walls-of-lucca/
Siena is another favored choice, and Fiesole is just minutes by bus to its perch high above Florence. (https://www.visitflorence.com/fiesole/).
Rome? Always a winner and one I can't imagine ever tiring of: oodles to take in there. No need to endure the mob at, say, the Vatican Museums unless it's a must for you and I wouldn't recommend it if you only have a couple of days anyway. Galleria Borghese has a great collection, excellent crowd control and only takes 2 hours to cover: pre-reservations are mandatory. You might like a ramble through the large park it's located in (Villa Borghese) as well. One of our personal highlights has been a looooong hike on the Appia Antica, and that one can also be biked.
http://www.parcoappiaantica.it
Anyway, I'd split your time between the two and do some reading to choose which one has the most appeal for allocating more versus fewer of your days. Give your interests, I'd probably go directly to Florence on Sunday, move to Rome early on Thursday (only 90 minutes by fast train) and then off to Civitavecchia on Saturday to catch your ship.