Florence, Cinque Terre and Amalfi Coast

Sure, look at "open-jaw" or "multi city" flights to maximize your time. Florence and Naples would be the ideal airports. If you want to swim, put the place you wish to swim last for better chance of warmer sea temps.
Transport via trenitalia, with bus, ferry, or private driver for the Amalfi coast.

The Amalfi Coast is almost 200 miles south of Rome. Cinque Terre is northwest of LaSpezia and Pisa--West of Florence and far north of Rome.
In two weeks, Rome, Florence and Venice are such important cities to see. And Cinque Terre would be secondary to them.
The trains are how you would travel from city to city--fast, efficient and inexpensive.

Would definitely help your schedule and rushed feeling if you flew into Pisa and out of Naples or vice versa.
Visit Cinque Terre, then Florence and then Amalfi Coast.
If you have 2 full weeks you definitely have time for all 3 if those are the places you want to see.
Probably 2 nights CT, 4 nights Florence and 6 nights AC is how I would break up 12 nights but giving another night to CT also works especially if your flight comes in late to Pisa or will be tired after long day of travel at the start a 3rd night should be added.
In general I usually advise against CT and AC on the same trip , why travel all that distance to reach another location with a similar feel and similar things to do?

I like both areas so am not dead set against it, if I had to pick one as long as the timing is not tight I would choose AC every time and the shear size difference and amount of things to do is why in my breakdown above so much more time is given to it.

I don’t know where your home airport is, but, if possible, I would start at Amalfi Coast (fly into NAP) and end in Florence (fly out of FLR).
The reason why you should end your trip in Florence is because your return flight will be early in the morning, and the Florence Airport is only 10-15 min from the city center.
If you leave Cinque Terre for last, the closest Airports (Genoa or Pisa) are each 1.5+ hours away.
If you leave Amalfi Coast for last, you are also just as far (1.5 hours) from the closest Airport (NAP).
In either of these last two cases, that would force you to move and spend the night closer to the airport on the last night.
From the Amalfi Coast to the Cinque Terre take a train from Salerno to Cinque Terre town of your choice. There will be more than one train change involved and it will be a long train journey.
Check schedule here below:
www.trenitalia.com

Sarah, do you have a wish list of the sites you want to see in Florence and the Amalfi Coast? This would determine how much time to spend in each location.

Florence... is a city full of museums, art, architecture, shopping and fabulous food.

Tuscany...day trips for wine, quaint hilltop villages Spas and again food.

The Amalfi Coast has a wide range of places to explore. Do you want to visit Capri, Pompeii, Naples and the Archeological Museum? Or mainly
the 11 charming villages hugging the coast.

Answers to these, will help us help you with a reasonable itinerary.

Okay, after reading your comments (as well as additional research) I am now going to save Cinque Terre for another visit. So the trip will be Florence and then head south.

I think you will prefer that more relaxed pace, enjoy!

In terms of flying out of Naples from the Amalfi Coast with an early morning flight.

Florence would probably be easier from where you are staying but the Florence airport is tiny and only a fraction of the number of flights from Naples. Whenever I have researched I rarely see reasonable rates on flights to/from Florence ; usually Pisa, Rome, Naples are all cheaper with Rome often being best for connections/flight options.

Depending on your budget though, I don't think spending your last night near the Naples airport is required if you save AC for last.
I believe I paid 120 euro for a private driver which my hotel arranged for an early morning comfortable and scenic ride from Positano to the Naples Airport ; if you stayed in Sorrento that would cost even less.