5/23-24 Naples (2 nights)
(Tour Naples on first day and Pompell (and maybe Mt. Vesuvius) on the second day)
5/25 Sorrento
5/26-27 Capri
5/28 Positano
5/29-30 Amalfi & Ravello (spend both nights in Ravello)
5/31 Travel from Ravello to Florence (I’ve already figured out how to do this; it’s about 8 hours of traveling)
6/1 - 6/4 Florence
6/5 Siena
6/6 Voltera & San Gimignano (spend the night in Voltera)
6/7 - 6/9 Venice
Leave on 6/10 - depart from Venice
I would like to add one more day in Venice. What do you think? Where would you recommend that I cut?
This is a lot of moving around! As suggested above, you can make a few base camps and save yourself a lot of time and probably money. Upon arrival the 23rd, I would go directly to Sorrento and stay there 4 nights. Day trip each to Pompeii, Naples and Capri. FWIW Capri is very expensive and somewhat overrated. Your precious time is better spent elsewhere than 2 nights on Capri, IMO.
Move to Positano or Ravello and spend 3 nights/2 days seeing both and the Amalfi Coast. Paestum is a stretch, and although I know my friend Zoe loves it, we were underwhelmed. I just do not think it is worth going all that way for. Again, there's so much else you could do/see first. Make a "next trip" stop.
That's the 23rd-29th, 7 nights. On the 30th move to Firenze for your 4 nights there (30th, 1st, 2nd, 3rd) and see Siena as a day trip. Volterra for 2 nights, 4th & 5th (car needed), leaving 4 nights for Venice.
Tha only place you really should have a car is Volterra for purposes of seeing some Tuscan towns. See San Gimignano on your way to Volterra. You could spend the next day there driving to/from Lucca and taking in the scenery. Skip Pisa.