I advise folks to get to their farthest point immediately upon arrival from overseas, yet I've only done it once, our first trip to Italy where we did Chicago-->Rome, then took a flight to Catania, Sicily. Didn't know what real jet lag was at the time but comparing it to subsequent trips I was no worse off than normal.
That being said, I'd start in Rome (agreeing that the Amalfi Coast is too far/too much this trip), providing your flight from the US to Venice arrives at a reasonable hour. Fly into Roma Fiumicino (1+ hours) or you could get over to Venezia Santa Lucia station and take a Freccia train 3 hours, 33 min to Roma Termini. Pros & cons to both--although I love trains and the ride to Rome would be restful, maybe it's better off you just schedule a quick one-way flight and be there quicker. Get your bags, taxi to your hotel, you're where you want to be!
And here's how I'd do your trip--
4/29 Rome
4/30 Rome
5/1 Rome
5/2 Rome
5/3 Orvieto by train (1 hr 17 min)
5/4 Orvieto
5/5 Florence by train (2 hrs 27 min)
5/6 Florence
5/7 Florence
5/8 Venice by train (2 hrs 5 min)
5/9 Venice
5/10 Venice
5/11 Venice to US
I usually say less stops are better, but if you get to Rome immediately upon arrival, you can fit in a couple nights in the wonderful Umbrian hilltown of Orvieto. Charming place, spectacular cathedral, awesome food, most importantly it's unlike any of the other three destinations. And it's on the way.
Logistically, these train rides are quick, efficient, comfortable and if you set it up in advance--like now--you can probably save at least 40% of the cost.
For a first trip, this to me is the best use of your time. You say you're not an art buff, but I said that too. And I defy you to visit Rome & Florence, which are so different on so many levels, and not have the art and the history behind it move you. While 3-4 nights apiece in these classic cities is not nearly enough, it's a taste. You'll be back.
Enjoy your planning!