Dublin? Paid - Croke park and/or aviva stadium, Guiness brewery. Just countless things to do really. So many wonderful pubs, cafe's, bars etc. Just wonder around the city. I'd highly suggest a bus tour out if you have a few days. So much stuff to see around the country. The likes of the ring of kerry, dingle peninsula or cliffs of moher, newgrange (5500 year old pyramid style structure.)
Free - Dail Eireann + a spate of free museums together, Trinity College, Dublin castle.
The economy is doing well right now so hotels will be expensive. Try hostel shining dublin, for very cheap rates if that suits you.
You'll find it hard to see most of many of these cities in a few days. I would have planned maybe at least 5 days in each, or split across numerous trips. European cities have a lot more depth than American cities if your interest is in culture or history. You could end uo looking at all the wrong things and have a dull trip to any one city because the trip is so short, and you may miss all the good stuff in an unlucky way. It does sound amazing though! Stock up on lucosade and drink plenty of Guiness/Beamish/Murphys and you'll be fine ;)