For (just) off the beaten track activities go to the Burren in County Clare. Tons to see and do there. The ring of Kerry (if you have good weather) is spectacular. Try biking it! Killarney is fun. Galway is a nice town. You must go to the Arran Island - take ferry from Galway. Great place.
Some excellent surfing off the west coast - but the h20 is very cold.
Be prepare for rain, and plenty of it. It may never happen of course
Way off the beaten track, but lots of fun. Go to a Gaelic football or hurling game. Have a look at the schedules at:
http://www.footballchampionship.gaa.ie/fixtures/ and do some googling to find out where the various stadia are. Any game with Dublin and/or Kerry will be mad. Have a few pints beforehand and afterwards and you'll probably have a fantastic experience. Oh, read the rules of the sports (summarized as: beat opponents up) on
http://gaa.ie/page/all_about_football.html so you won't have to pester semi-sober locals during said games.
Social decorum, or how not to get your beaten up (at worst) or merely avoided (at best):
Please, please, please never wear "tourist green", and [if male] don't wear trousers in any color other than black or blue denim. Don't tip. They'll think you're completely mad, or worse, a tourist.
Oh and watch out for "tourist heads" i.e. pints poured with unnecessary inches of head. Locals would never put up with it, neither should you. Guinness should have 0.5 inch head not 2-3 inches as they try to serve unsuspecting tourists (see green and trouser comments above).
Enjoy!