Please don't just dismiss Sonos as more costly and less effective. It may actually less costly and more effective! Especially depending on the speaker(s) the OP wishes to purchase. Add in the Airport Express or AppleTV on top of speakers and it could be more costly than a Sonos setup. Admittedly, Sonos can get very costly, but can also be as little as $200 for a single speaker or $350 for a pair.
Also, "For only music in another room" is exactly what Sonos is absolutely brilliant at offering. In fact, playing selections throughout a home, either the same selection to a single or multiple zones, or different selections to a single or multiple zones, is part of its attraction.
While you could use Sonos to play from a running Mac with iTunes, you can also play from many many other sources, including from your iPhone, iPad, Android phone, etc.
I am not pushing Sonos. Airplay likely would work perfectly well for the OP. But Sonos is something to consider.
PS - I use both Sonos and Airplay for different purposes. Sonos to a couple of rooms inside, and Airplay to my outside speakers.