Can you store some of your bags for several days at Florence train station..Heading to Cinque Terre..
Then coming back to day trip from Florence for a few days..
Yes, but it is costly. €6 for the first 5 hours, €1 for each additional hour and that is AFAIK, per bag. Your best bet is to pack light enough that you don’t need to shed baggage to take that short trip to the Cinque Terre.
Radical Storage (formerly Bagbnb) has several locations near Florence SMN station at 5 EUR/day.
Or check with the hotel (if you will be staying at a hotel) in Florence where you will eventually be staying and see if you could leave your bags there and retrieve them when you eventually check in. Very possibly there would not be a charge to do so. The cost of taking a quick taxi to the hotel and then back to the train station might be far less than bag storage at the station. Of course, this advice is irrelevant if you are staying in an apartment.
You can contact the Florence train station and ask how much it will cost at https://www.italiarail.com/contact-us.