OP I've done similar to your intended trip, also on my own and it was hectic in places, but overall very easy after the first day getting my bearings.
Taxi to from airport to hotel is the best way, it's a shortish journey.
How long are you planning to travel for? Going during school terms is much cheaper and easier to get around - if you can.
Just a flavour of what I did -
Pisa one night
Lucca two nights (fabulous place and not to be missed.)
Florence two nights. It was heaving, and although it was great to see, I was a bit overwhelmed by the crowds and the queues and that was in February during school term!
Bologna two nights. Fabulous, great quirky vibe and the FOOD! Came home from the airport there.
I used trains all the way, some pre booked, some bought at the station. Trenitalia as mentioned is a good app.
I know my itinerary doesn't include Pompeii or Rome, that is my next adventure. But I think I'll do Herculaneum and Ostia Antica, the road less travelled as they say!
Sounds amazing, have a ball.