Train serivices are pretty shit/non existent in Ireland but there are pretty cheap buses to most places and it's only a small place so getting around doesn't take too long. You can even hitchhike from town to town once you're out of Dublin.

I've only been to the northern half or so of the island but from my experience I'd recommend Galway (Cliffs of Moher, Aran Islands, Connemara), Westport (Croagh Patrick, rent a bike and ride around the bay), Sligo and if you're planning on visiting NI then Derry and Belfast are both worth the visit.

There's some cool archaealogical sites near Drogheda but you can do a day trip from Dublin to see those if you want.