Killarney National Park is located along the famous Ring of Kerry. It is not a very large park, but has some of Ireland's best to offer.
Not only will you find hiking trails that, despite the sometimes crowded park, will give you a feel of being far from the modern world, you can also visit the historic Muckross House.
Entry and parking is free (but make sure you get there before 11am, or all spaces might be taken), but a visit to the house will require an admittance fee (around 5 euros).
For the hiking trails, there are several options. Some are marked with numbered signs (though you have to pay attention as they might be half-hidden in the grass); for others, a trail guide will come in handy. The Lonely Planet guide Hiking in Ireland has several trails in and near Killarney National Park. Trails go through ancient oak forests, along cliffs, and past Muckross Lake and Lough Leane. You will be rewarded with amazing views, but beware of the bitemes that may be out in shaded, humid areas.
For those who feel hiking is too much effort, you can also hire a jaunting car for a tour of the House's grounds. The driver will even take you into the forests up to the point where Muckross Lake and Lough Leane meet. A tour should start at about 30 euros.