From Seattle, you may not have any nonstops to Italy, but that doesn't mean you can fly through London, Amsterdam, or Paris to Italy all on one ticket. That's the best way to go; otherwise, you waste a lot of time, as well as introducing risk. If you buy Seattle to Italy all on one ticket, the airline is responsible for getting you to your final destination, at their expense. If you (for instance) buy Seattle to Paris and then separate tickets (train or flight) to Italy, and your flight from Seattle to Paris is delayed so you miss your next leg, YOU are responsible, both logistically (you have to figure it all out on your own) and financially (you have to buy new tickets at last minute prices).
Furthermore, you can book "open jaw" (also called "multi city" or "multiple destinations") tickets. This means you can, for instance, fly Seattle to Venice and Naples to Seattle, all on one ticket. These are usually not much more money than a simple round trip, and even if they do cost a bit more, you save time, money, and hassle by not backtracking.
So, once you've figured out where you want to go in Italy, you can start looking for multi city flights from Seattle to Italy. Of course, if you really want to see London, Paris, or Amsterdam, you can do that as one of your legs. But, flying somewhere you don't want to see, just because they have a nonstop flight, is a false economy.
I also want to emphasize that you need to stay in the areas you want to see, not several hours away from them. Trying to see Florence or Venice from Milan is like trying to see Seattle from Portland.
Your idea of home bases is good, but use a narrower radius. For instance, from Venice you can see Verona, Padova, Ferrara, and other places not too far away; just don't try to see Florence or Milan from there.
Depending on how long you have, you'll probably have to cut down your wish list for your trip (you've already cut Sicily, but be prepared to cut more). When it comes to travel, less is more. Stay in fewer places and spend more time actually seeing them. This also reduces cost.
Added bonus: by staying in fewer places and just seeing things in the region, the transit becomes much easier!