if you want to fly from a local airport there's not much choice - to dublin from newcastle or leeds with ryanair, and that's it.
http://www.ryanair.com/en
or, travel from almost anywhere in the uk to dublin by train and ferry, one way fare starts from around £38.00
http://www.seat61.com/Ireland.htm
a similar fare is obtainable for a coach/ferry journey, if booked well in advance.
otherwise, take your car - via holyhead journey (from newcastle) comprises 270ml of driving