Day Trips -
County Wicklow - Grab a tourist bus and visit the Avoca headquarters, or take Dublin bus to Enniskerry and explore the Powerscourt House grounds. No need to pay the admission fee to walk the beautiful park approach to the house.
The Wicklow Way trail runs from Marlay Park in Dublin to Clonegal, a week-long hike divisible into shorter sections.
See Wicklow Way Bus for a shuttle service to facilitate a day-hike.
You might want to make a longer trip to Killarney, but if you take a train early and come back late, you could do Killarney as a day trip.
Go biking in the national park- it’s one of the most beautiful places I’ve ever seen. Bike rental shops are along the main street, and they’ll explain your route options. Don’t miss Ross Castle!
Longer Trips -
Sligo - Yeats county, beautiful and wild. Might be difficult to get around without a car or a tour bus. Also visit the Neolithic tombs.
The Aran Islands - You need at least an overnight stay if you’re coming from Dublin; ferries leave Galway in the morning and come back in the evening, so you have to start in Galway the morning of your trip.
Hostels are cheap, and you can easily explore Galway the evening you arrive. On the Aran Islands, rent bikes and ride around.
This is the stereotypical Ireland you’ve fantasized about, and one of the most solid bastions of the Irish language. The other most beautiful place I saw in Ireland (the first is Killarney). The cliffs off the island of Inis Mór compare well with the Cliffs of Moher, so although the tourism gods might curse you, I think you should skip the Cliffs of Moher and just do the Aran Islands if you only have time for one. On Inis Mór, definitely hike up to the prehistoric fort Dún Aonghasa, and definitely send your mom a knitting pattern from the island; the famous Irish clan patterns originated in the Aran Islands, although whether the clan patterns are authentic is in doubt.
Photo Gallery and Additional Commentary here:
PHOTOS - An American student's guide to the best tourist spots in Dublin slideshow