I agree with @ukbea. I’ve been three times, and I certainly think it’s smart to pick bases. Florence is a fantastic base, and you can visit Siena, Verona, Cortona, even Venice from here. Though Venice is fabulous, and I’d recommend staying there as well.
Positano is magic. Like, hands down, one of my favorite places in the world. It takes a bit to get to, so I don’t know that I’d advise it for a ten-day trip unless you went straight from Rome there, but at some point, go. So worth it.
Keep in mind your travel days, even though train travel is good, it takes a bit. Travel days end up being kind of a wash for me.
Day 1: So depending on what time you fly in, find where you’re staying, check in, rest a bit, you may or may not have time to explore.
Days 2-4: Depends on what you want to see in Rome. Vatican alone can take up an entire day. Do you want to see the Colosseum, Pantheon, Spanish Steps? Do you want to visit Pompeii? That’s a day trip.
Day 5: Travel to Florence. Check in, etc. Walk around, check out the markets.
Day 6: Duomo, lunch, Santa Maria Novella (great old pharmacy turned to perfumes and lotions, if you like that sort of thing)
Day 7: Boboli Gardens (beautiful) with half day trip somewhere
Day 8: Day trip
Day 9: Return to Rome if you fly out?
Day 10: leave
You can see why you wouldn’t want to switch accommodations more than that based on this. Another idea may be to stick to a region. Rome, Pompeii, Naples, Sorrento, Positano. Or fly into Milan and do Venice, Cinque Terre, etc.
Hope this helps! I adore Italy and can’t wait to go back again.