Just so there's no confusion, if you depart from LAX on April 14, you will arrive in Italy on April 15.
The My Way tour as described on this website includes two nights in Florence and makes a stop in an unspecified Tuscan hill town on the way to Assisi, which is in Umbria. So the tour does provide some time in Tuscany.
Orvieto is in Umbria, not Tuscany; if you're looking for a base from which you can see more of Tuscany, Orvieto is not the right choice. I like Orvieto a lot; for a fairly small place, it has a lot of sights. Although it's popular, the pace of life is slower than in Rome, making it a pleasant place to recover from jetlag.
Different people will have different opinions about whether you should spend your first night in Rome or travel on to Orvieto upon arrival in Italy. It's rather a hassle to check into a hotel for just one night, and it's entirely possible you'll be barely awake on your arrival day, anyway. Therefore, I don't think there's a lot to be gained by spending Night 1 in Rome. If you want extra time in Rome (and you would certainly benefit from it), I think it makes a lot more sense to add a day or two after the tour, ideally staying on in the tour hotel so you avoid an additional hotel change.
My choice would be to head to Orvieto. You can wait until you get to Italy to buy the ticket so you don't have to worry about a major airline delay causing you to miss the validity period for a pre-purchased regional train ticket.
I'd take the train from Orvieto to Venice. If you want to fly, you'd probably do better by returning to Rome rather than going to Florence, which I think will take longer by the time you figure the short trip from the city to the (small) airport. Even going to Rome, I'm not sure flying would save much time at all by the time you go through all the necessary connecting steps. For me the train would be preferable because there's less hassle.
Part of the Orvieto-Venice trip will probably be on a Frecciarossa train, though there's at least one direct InterCity departure. The Frecciarossa part of the ticket will be considerably less expensive if purchased well in advance, but that will tie you to a particular train on your chosen day. I think InterCity tickets are also price-variable, but I'm not absolutely sure about that.