Nikhil, on the weather, the best is to look up each major city with a Google search. That way, you can determine if a given temperature is desirable for you.
In terms of an itinerary, I just came back from a one-month trip to Italy and went to similar cities. I'm happy to share that with you. I'm not sure if this site allows me to share my travel blog where I posted it as I'm happy to share my planning tools. It's my first name then Italy dot com. Or messenger me in Facebook.
I divided my trip into thirds: North/Tuscany, South/Amalfi, and Rome. Since you only have 10 days, I would do 2 basecamps. Then, you anchor in two cities and take day trips to other towns.
Based on what you mentioned, I would anchor in Florence and then visit Cinque Terre and Lucca instead of Pisa...or both Lucca and Pisa in one day. We went to Venice from Florence (via train) and just took a backpack to stay over for one night in Venice.
Then Rome! Fabulous experiences.
We flew in and out of Rome (for the best prices) then took a train up to Florence immediately and ended in Rome.
I hope that helps! You can plan this! Get air tickets first, then Airbnb/Vrbo or hotel. (we loved our Airbnbs). Then you can fill in the days with fun!