Iceland to Ireland or Vice Versa

I did exactly this is August.

Both Icelandair and Play Airlines flies from the US to Iceland and then from Iceland to Dublin. You would need to take a train to Belfast, or a direct Bus from the aiport, as no US or Icelandic airline flies there.

To go back, you would have to go back to Dublin and then fly through Iceland to get back to the states.

Depending on where you are in the US, Delta and United also fly to both Iceland and Dublin. You could then take either Icelandair or Play to Dublin. So, DL or UA to Iceland, Icelandair or Play to Dublin, then DL or UA back to the US.

Icelandair lets you stop in Iceland, up to a week I think, with no additional fare. I expect you could do that going in either direction. They also have various hotel, tour, and car packages that might be worth exploring. Whether you want to visit Iceland before or after Ireland depends on the season and how you handle jet lag, i.e. whether you want to mitigate the time change going east or going west. (I have more trouble adjusting after returning home, so I might choose Iceland on the way back, especially if I was traveling in spring when Iceland might be a little warmer on the way home.)

Choosing Icelandair forces you to transfer in Reykjavik (KEF) going both ways, and you might prefer a nonstop one way if your home airport has one. But that would likely cost you more than a round-trip ticket (with stopover) on just one airline. And KEF layovers are pretty easy and efficient.

The Aircoach bus between DUB and central Belfast is quick, comfortable, cheap, and covenient, so you could use it at the start or end of your Ireland visit instead of having to go into Dublin twice. You could fly into DUB, take Aircoach to either city, then train between them, then Aircoach back to DUB for departure.