For about three centuries, Galway was run by fourteen merchant families, the Tribes, who traded wine, salt and fish with Spain and France and built themselves stone town houses inside the walls, while the Gaelic Irish outside were kept out of the city by law. The Parliamentarian army ended that with a long siege in 1652, the trade moved elsewhere, and Galway slept for a long time on the edge of Europe. The Great Famine of the 1840s hit the west hardest of all: Connemara and the Claddagh emptied, and emigrant ships left from Galway's docks. You still see the merchant city in its lanes, its old church and the stone arch by the river.
Today it's a small, young university city of about 85,000 people, and probably the most fun town in Ireland: buskers on every corner, a festival most months, traditional music in the pubs every night, and the Irish language spoken in the villages a few kilometers west. It's also the gateway to the wildest part of the country, so five days here is really three in the city and two at the edge of the Atlantic.
This plan keeps each day simple. Day one is the medieval Latin Quarter, day two the cathedral, the Claddagh and the seaside promenade at Salthill, day three the Cliffs of Moher and the Burren, day four the island of Inis Mór, and day five the Saturday market, the river and a farewell night. The plan works from a Tuesday, because the city museum is closed on Mondays (and on Sundays outside summer) and the market is on Saturday. For the bigger picture, see our page on Ireland.