Since you already have pre-purchased train tickets to La Spezia and hotel reservations, I'd probably just carry on as planned. As someone else mentioned, it's difficult to accurately predict what the weather will be like in 10 days. If the weather is good, you can at least have a look at the five towns, although they will be "quiet" and some of the businesses may have closed for the season. The two trail segments that were open this year may be closed, so I wouldn't count on that. If the weather isn't good, you can resort to "Plan B" and explore Florence or wherever.
Did you purchase the train tickets to connect with an inbound flight to Milan? That's generally not recommended here but since it's done you'll have to take your chances. Keep in mind that if you're travelling on one of the fast trains, the ticket will be specific to train, date and departure time so can only be used on that one train. If you board another train by mistake, you'll be subject to hefty fines which will be collected on the spot!
If you use Regionale trains, Metro or Buses in the city and you buy tickets locally, these must be validated prior to boarding on the day of travel or again, hefty fines collected on the spot!
As you're planning to rent a car, be sure that each driver listed on the rental form has the compulsory International Driver's Permit. You'll also have to be vigilant to avoid the dreaded ZTL areas (hefty fines for each pass through one!), speed cameras, etc. Do NOT drive in Florence!!!
As you're planning to travel by train from Naples to Sorrento, you'll be using the Circumvesuviana commuter train, which has a reputation as a haven for pickpockets and scammers. Be vigilant and wear money belts! You may find this recent thread interesting.....
https://community.ricksteves.com/travel-forum/italy/circumvesuviana-train-was-a-terrible-experience