I've spent almost 6 months in the Naples area.
While it is possible to do the items on your list, here are my impressions. Keep in mind this would have 2 very long days.
Day 1. 8-9am departure from Florence. Arrive between 11-12 hotel by the train station, or this cute "little" BnB. Good bus &metro connections to the rest of the city.
http://www.booking.com/Share-PsPW7qw
In the Centro Storico. Short 5 min walk to the Archeological Museum and metro (line 1). 10 min ride to the train station. All the famous pizza places 5-15m walk.
Today, visit either/both Capella Sansevero €7 pay out of pocked for this site. or Castel Dell Ovo. Free.
Look carefully at this. For the sites you wish to see, it will save you money.
http://www.campaniartecard.it/artecard/en/content/luoghi
It's a 3 day pass which includes 2 free sites, use the most expensive first, with discounts to others. Transportation on metro, busses funiculars and regional trains. If you see Pompeii, Archeological Museum and Ercolano, it pays for itself. Not including transportation you'll use.
Day 2. Pompeii Scavi 38 min. allow 3-4 hours,or 6, a guide here is a good plan and then Archeological Museum.
Day 3. Ercolano Scavi. 20 min. To me, it's not worth a 2 hr 15 min bus ride 1:35 to, 40 back,to Naples to spend an hour on the summit.
My daughter and granddaughter have lived here 15 years and just visited last year. If you do decide to do this, you need to walk (30 min) or taxi (5 min) to the Pompei station to board the bus to Vesuvius. €10.
So, 2 choices here, head back to Naples and see some other sites, or continue to Sorrento and and maybe Positano. 1 hour bus between the two. Then Circumvesuvian back to Naples. 1hr.
Dependent on adventure 8-10 hr day.
Day 4. Regional train to Salerno, 43m, ferry to Amalfi 35 min. Towns along the way. Maiori, Minori, (stop at the fab pastry shop Sal de Riso in Minori). Arrive Amalfi, bus 30 m to Ravello. Villa Climborne and Rufolo are beautiful (discounted) return to Amalfi, 1-2 hrs then bus
back to Sorrento. Circumvesuvian to Naples.
8-10 hour day.
Timetables for transportation in the area.
https://www.sorrentoinsider.com
Edit: ferry schedule for Salerno to Positano.
https://www.travelmar.it/en/hours
Sita bus on the AC is not included in the arte card, scroll down to see prices.
Prices on some of the sites I've enjoyed with prices and discounts. All prices are in Euro.
Pompeii Scavi. €15/9,00.
Arch. Museum. 12/6,00.
Ercolano. 11/5,50.
MAV, typical 3-D. $$ better spent on souvenir book.
Naples Soltternea. 10/5,00.
Palazzo Reale. 6/3,00.
Castello dell Ovo. Free.
Castel Sant' Elmo. 5/2,50 Combines well with...via funicular.
Capodimonte. 12/8,00.
Caserta Palace. 3 tiers.
Palace & Garden. 12/6,00.
Garden only 8/4,50.
Palace only. 9/4,50.
Villas Climborne Rufolo 11/7,00.
Hope this is of some help.