Affordable Flights from Seattle to Rome?

It's a bit early to book for January 2019, but you can get a good idea of which carriers serve the route and how much they charge by searching on Kayak.com. Personally I don't know how many fly Seattle-Rome nonstop -- if any -- but you may have a few choices of where to transfer if it is a 1-stop (as opposed to nonstop) flight.

When you go to book, I suggest writing down your desired flights and then closing your browser and going to the actual airline website(s). Sometimes you can be booking through a 3rd party without knowing it, and this causes hassles later if you need to contact the airline for any reason.

Nonstop flights from Seattle to Europe go to Reykjavik, London, Amsterdam, Paris, or Frankfurt (none to Italy) on Delta/KLM/AirFrance, British/Iberia/Vueling, Lufthansa/United, Condor, Norwegian, or IcelandAir.

In the direction toward Europe, you could plan a budget flight connection on Easy Jet or other independent budget carrier, but confirm that it leaves from the same airport, pack light and plan to carry on bags so you don't have to collect and re-check them, and be prepared to buy a new onward ticket if you miss the second flight. In the homebound direction, this is much more risky; e.g., you risk missing and having to re-purchase a very expensive trans-Atlantic flight. So only do that if you have at least an overnight layover at the connection city.

Do some searches on Google flights and set a fare watch. This will give you an idea of the going rates and how much they vary. Iceland air has Seattle flights and often very good sales. Also don't forget to consider hassle factors that are not price oriented - I am willing to pay an amount extra not to have a 24 hour travel day with multiple stops. As was noted switching carriers half way comes with potential down side.

Definitely do some searches on Google Flights and set up fare watches.

I flew roundtrip from Seattle to Rome during March 2017 on American Airlines for $719 - if I would have waited a week it would have dropped to $551. I just flew British Airlines this month from Seattle-Amsterdam and then from Rome-Seattle for $619. Both fares included meals, seat reservations, one checked bag, and each roundtrip was on a single reservation so that if a flight was missed because of a delay the airline was responsible for getting me to my final destination. I purchased the fares about 2 1/2 months before the departure date. I did have some flexibility regarding travel dates and that helps. I keep a close watch on fares and when a good one comes up I jump on it. Since you are traveling in January I would expect you should be able to get some good fares.

Aerlingus will be starting nonstop flights from Seattle to Dublin with connections to Rome. They probably don’t have flights listed for January 2019 yet. One nice aspect of Aerlingus is that you can go through US customs and imigration in Dublin on the way home.