Ireland on a budget is best done (imho) with An Óige.’
This is the Irish youth hostel foundation http://www.irelandyha.org/
As for where to go? Kerry Kerry Kerry and Dublin Dublin Dublin 
Actually I probably have to throw in a Galway Galway Galway there too.
Get one of the rambler packages from An Óige, it combines a buss/train-pass with sleepovers in the hostels. Promise me (and yourself) that you will go to the Black Valley in Kerry (its in the whole Dingle area) and also you should check out Valentia Island, site of the first ever transatlantic cable! grin It really is beautiful, there is a lot to see, and the hostel is actually old coastguard houses.
I could babbel on forever, but my own suggestions would be to go for the following things in the following regions, disclaimer, I am not saying that the other areas do not also have these qualities in abundance!
In kerry, do the nature thang, walk on the cliffs, swim in the atlantic with the dolfins, put yourself in perspective with the world. Eat and drink well.
In galway do the whole pub scene, it is a young and vibrant town, but obviously with its history intact and everywhere, this is the place to go to Irish bars and have a jar and hear some real irish music, as opposed to the Oirish tourist traps in Dublin. Eat and drink well here too. If the timing is right, enjoy the Guiness Oyster Festival, gDAMN 
Dublin is a major european city, just on a small scale
Skip the “irish” pubs here, and experience some of the nicer upscale bars to balance out your experiences with the culchies grin…
I am gonna hang for that… Uh, anyway, em… yeah where was I …
In Dublin you should shop and party, but also do the “historical education” thing. You can do this with a minimum of effort as Dublin itsself is so small. Do check out the tourist standards, they are famous for a reason, the GPO, Trinity College, the Book of Kells. A slightly less famous suggestion, but not without entertainment or educational value is to visit Glasnevin Cemetary. There are buried some of the most famous people in Irish history and you can also see one of the old “monk towers” from when the Vikings used to attack (large tower, door half way up, bring the ladder with you on the way in, frustrated Vikings).
Its a bit of a cliche but you really should check out The Brazen Head. It is the oldest pub in Ireland, if not the whole of europe. Just take the one, and then shag off to the other side of the river to the Morrison hotel and have a civilised drink there 
My one true tip to all visitors to Ireland, especially from the states is this. Please be careful of yourselves and your belongings. Try not to stick out as tourists (plaid trousers, kiss me Im irish t-shirts, flat cap and camera). We are not some race of benevolant little leprechauns as so many tourists seem to think, and if you leave yourself open to it you will get robbed, just as surely as in Paris, London or New York. Dublin has a fairly huge drug problem, and that video camera you have slung around your shoulder will really make somones week. Put it away till you are using it for gods sake. </endrant>
I only say it cos I want you to have fun, and take only good memories back with you
Be smart, be safe, have a fucking ball 