Hi Karlye -
To echo some of the others, your itinerary will have you spending too much time checking in and out of hotels and dealing with transport that will eat into your sightseeing. No, you don't have "all day" in Florence, Venice or Siena (you'd have so little time in the latter that it wouldn't be worth the trip) but only partials, and in all cases you'll be having to deal with your luggage before heading off to have fun.
No, a car is definitely not advised for the CT. For weather reasons, I'd cut that one anyway and add the time to places you've shortchanged. I'd also look at going directly to Venice from Milan unless there's something there that's a must-see on your list. Day-trip Siena from Florence to eliminate another hotel move.
Your Rome dates are off by a day: Saturday is the 28th and not the 29th
Flying into Milan on Thursday 3/19 am- train to Venice
Friday 3/20- Venice
Saturday 3/21- Venice (or train to Florence)
Sunday 3/22 - train to Florence
Monday 3/23- Day trip from Florence to Siena by train or bus
Tuesday 3/24- Florence
Wednesday 3/25- train to Rome
Thursday 3/26 - Rome
Friday 3/27 - Rome
Leave Saturday 3/28
Whether to spend two full days in Venice or move to Florence on the 21st is up to how much in Florence you're interested in? There's a lot to see there if what's on offer trips your trigger. Do consider that the National museums (Uffizi, Accademia, etc.) are closed on Mondays, if some of those are on your list. Just 2 full days is pretty short for Rome but again, it's up to how much of it you're interested in seeing.