SEA-AHO
Most airlines airlines flying to Alghero are low cost European airlines, such as RyanAir, Easyjet, Volotea, etc.
Since that is a summer destination, most of those flights are seasonal too, however on the 1st of Sept most should operate.
The only traditional airline that flies to AHO is Alitalia from Rome FCO or Milan LIN (Alitalia is a SkyTeam alliance partner, like Delta)
So you really have these options to fly from SEA to AHO:
1. You select a Delta/Alitalia combination that takes you to Rome FCO first, then to AHO, such as SEA>AMS>FCO>AHO. That is probably the only chance to do it on one ticket.
2. You go to Italy first, then you go to AHO maybe the next day, on a separate ticket with a low cost airline. In this case you wouldn't go to Rome, because no low cost airlines fly from Rome to Alghero (only Alitalia, see no. 1). RyanAir flies from Pisa, Bologna, Bergamo. Easyjet from Milan Malpensa, Volotea from Venice and so on (for a list of airlines flying to AHO see Alghero Airport on Wikipedia.
Skyscanner is a good aggregator for low cost flights. Or you can go to the individual low cost airline websites.