If you happen to have relatives in Ireland, the best reason to travel to Ireland is to experience personal family history. However, if you don't have the luxury of having Irish relatives and are traveling on your own, you might consider other ways of immersing yourself in the culture.
Despite whatever the songs might suggest or whatever belief you have about the Irish, most passersby on the street are not necessarily the friendliest of people. You may get lucky and run into some friendly folk, but during the daytime it seems that everyone has their face to the ground and is in a hurry to get somewhere. You might even find the crowded walkways of the city to be pretty pushy at times (and remember that people don't generally form lines in Europe).
Certainly see the sights, but when visiting a large city like Dublin or Limerick, the best thing to do is to experience the night life. In America, the term "night life" often refers to a crowd of people in their twenties of early thirties hitting the clubs to dance. Ireland's night life seems to be a bit different. For instance, while many people seem unresponsive during the day, crowds poor into town hitting pub after pub at night and are eager and friendly to talk to new people. Remember, there are 90-some pubs in Dublin alone, so don't be satisfied with the first place you stop at. Hit a few pubs (where there's room to get in). There is typically live music at most pubs or taverns, so find a band you like, order a Guinness and talk to strangers. If you happen to play an instrument and you can find a bar that doesn't have live music, you can do what is called a "session" where you sit down and start playing. Others with instruments may randomly join you, and more often than not you'll have locals singing along, clapping and buying you rounds.
The tourist attractions at the major cities can be pretty interesting. The Waterford Crystal factory in Limerick is a must see, as is the Kilmainham Goal and the Jameson Distillery in Dublin. The famous Guinness factory is a bit overrated and sometimes seems to be more of an advertisement than a tour. Still, you get a free glass of Guinness at the end.
When you're tired of visiting the cities, head out to the coast near Killarney and the Dingle Peninsula. Mind the beaten up, tiny roads. The drive alone can be an adventure. The Cliffs of Moher is another must see.
Overall, Ireland gets mixed reviews. On the one hand, it is very pretty. However, most of the history revolves around their famous revolution, which gets a bit monotomous; it's a myth that Irish has great beer (other than Guinness and maybe a couple of other beers, the locals favorite beer is Budweiser); and the towns are often run-down, dirty and poverty stricken. Some say that Scotland's a better visit for its natural beauty and England is a better trip for its culture and atmosphere. Still, Ireland is a unique experience and one new visitors won't soon forget. There's much to see in Ireland, so get in what you can and enjoy.