As per the posts above, the Burren and the Cliffs of Moher are the major draws in Clare. Visit Doolin if you can (it makes a good base for both of the above, and affords some good views of the cliffs from its harbour. Be warned, however, that it's a huge tourist draw in and of itself (it's a center of traditional music) and is thus crowded in the peak season.
If you have time (a few days, at least), visit the Aran Islands. Inis Mor features a spectacular ancient (megalithic, if I recall correctly) fort named Dun Aungus. It's a semi-circular stone fort perched on a 300ft cliff.
My advice would be to take the coast road as much as possible, detouring inland for significant locations. Connemara is well worth spending time in. Achill Island in Mayo will provide a day or two's exploring.
Sligo itself offers some very impressive scenery - the Ox Mountains are not insignificant, and have a wonderful monolithic feel to them. Particularly spectacular is Benbulbin just north of Sligo town. While you're in the area, visit Lough Gill just to the south and east of Sligo town.
Since you're so close, and if you have time it's well worth taking the drive to Slieve League in Donegal. It's a long drive from Sligo if you're just doing a day trip, so it's a good idea to get accommodation locally (somewhere around Killybegs is a good idea, but stay on the west side of town if you don't like the smell of fish). Slieve League is a spectacular mountain with the (purportedly) highest sea cliffs in Europe - although I've seen the same moniker given to cliffs in Scotland, Shetland and St. Kilda.
There's a (cringe-inducing in places) road to an overlook at the cliff tops that gives a very impressive view. Go at sunset for a real treat, although you may be eaten alive by midges. Buy midge repellant and you'll suffer less. Nonetheless, it's worth the trip.
Hope that helps.
Cheers,
Peter
Fine art prints:
www.petercox.ie
