Some of the charm has been lost due to its new wealth and foreign influence, but the farther you get into the country, the closer you get to the old ways.

I highly highly recommend Belfast and Giant's Causeway up in Northern Ireland and newgrange not far from Dublin in addition to the other suggestions above.  

Car hire is very cheap. You should spend at least a few days exploring. So many random megaliths and charming little villages to make just getting lost magical.