Go to O'Maille. The absolute best aran sweaters in the world, no ifs, ands, ors, or buts. Pricetag to match, but absolutely worth dropping into when you're in Galway. Even though I wasn't in the market for one when I was there, the proprietor was very friendly and happy to show off her wares, and enjoyed talking about them. They've done a tremendous amount of work to preserve the skills at the top end, training new knitters and providing an outlet for the best knitters to sell their goods at a higher price. I was shown a sweater knit by a women who the store owner said was one of the best ever, and it was shockingly good. Incredibly dense, and even the ribbing at the bottom was cable knit- in three different braid designs in a regular pattern.
Seriously, even if you're not in the market for a 350 euro sweater that could probably sub in for a parka on most occasions, you need to at least feel one.
There are also a bunch of touristy shops both in Galway and on the islands selling the much lighter weight frame loom knitted versions for ~50 euros or the lesser hand knits (which are still kick-ass sweaters) for ~100 euros, along with tweed caps all through there. Even the cheapest versions around are all wool and pretty nice sweaters in their own right, don't be afraid of them and don't go home without at least something. And get a tweed cap, but be careful and try them on first. Different makers have different cuts, and some of the fuller cuts just looked silly on me. I wound up getting one from John Hanley & Co. Hannah Hats is probably the biggest name, but their regular cut was just too full and their closer cut had a ridiculously long bill for me. Whatever you buy, it'll form to your head a little better after a shower or two.