Galway, in the county of the same name for its university life, bustling pedestrian-only zone and a bundle of restaurants and pubs. You can also catch the ferry to Inishmore at the dock, a short walk from downtown.
Doolin, County Clare - the Mecca for traditional Irish music in its four pubs, especially the granddaddy of them all, O'Connor's. Also, its ease of walkability and awesome location right on the coast. A smaller ferry departs the village to Inishmore and Inisheer.
Leenane, County Galway for the absolute prettiest setting in all of Ireland at the end of the country's only fjord and between three mountain ranges. It also features one of the finest restaurants in Ireland, The Blackberry: www.blackberryrestaurant.ie Reservations a good idea in the summer. Yummy lamb stew.
Glencolumbkille at the far southwestern end of County Donegal (due west of Galway Town) is perched right on the Atlantic and offers a hike or two to the top of the surrounding hills for dramatic views. The one pub in the village - Biddy's - hosts trad music sessions many nights in the summer. Threre may be just the one restaurant left where I ate when last there in 2007. Best to check ahead, as it's a drive from anywhere.
And all have wonderful B&Bs, though Glencolumbkille has only a handful.
You'll note that all four towns are in the West, where you'll find - by far - Ireland's best scenery and experience Irish culture, language, music, etc. much, much more than most anywhere else in the country. The first three towns are also in or quite near Connemara, Ireland's most scenic region.
Slainte!