Ciao! I would suggest you just use the train/rail system to head west from Florence and either pass through Pisa or Lucca and make a stop there, then you can continue north toward La Spezia. Once you get there, you can change trains and go toward Monterosso al Mare and that will take you through ALL of the five Cinque Terre, which everyone loves to visit to walk/hike. Do you like to hike? I would highly recommend it IF the temperatures are not too warm when you are here. Hiking footwear is recommended to go on the trails, do not attempt otherwise.
I would highly recommend you stay in one of the Cinque Terre, we personally really love Manarola for it's charm but also enjoy Vernazza and Monterosso al Mare. Then you can keep on heading north along the coast to enjoy other towns, including Genova (Genoa). Had you already decided where you want to end up along the Riviera?
