There's more advice in the thread "Looking for Irish Music" dating back to July 24.
The town of Corofin, south of Galway City, has an outstanding night of music and story telling one night a week. It might not start till the summer, though.

Doolin is the place to go for real Irish folk music. Much of it is impromtu music performed by amateurs.

Try Mannion's Bar in Balla, County Mayo...not far from Westport where Matt Malloy's is. On Tuesday nights there is a group called JoBange who plays at Mannions and they are definitely a local group and this pub is a very "down home local" pub in a small town and not touristy. Good people there...good Guinness! I'm headed back to Ireland in April with my friend...whoo hoo...can't wait!!!

Just got back from Ireland a few days ago and spent the last four nights in Galway. Live trad every night (except when soccer matches are on the TV's) at many pubs there. Usually two sets a night, one around 5pm and another around 930pm. We enjoyed Taaffes on Shop St. and Monroe's on Dominick (a block across the river). Monroe's also had live set dancing on Wednesday nights which was fun to watch. Nothing we found was touristy and the musicians were good but we were there in the off-off season.

However, even during the non-tourist time of year the pubs were packed, lots of college students. Since the pubs are so crowded the conversation level is very loud and will drown out the music unless you are in the room with the musicians. If you want to actually see the musicians you must get there early before the set starts.