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Hire a car or take the bus.
Bus- Bus Eirean, the trains are basicly OK.
Rent a car, be patient and stay on the right (which would be left) side of the road. And watch out for sheep!
We travelled from the continent to England with the North Sea ferries, drove across England to Holyhead and got the ferry to Dublin from there.
We had our own car with us which meant we were free to go wherever we wanted to go.
Just be careful of the traffic jams on some of the narrow roads - but eh who is in a hurry anyway?
Watch out, there are loads of sheep on the streets, especially in the 'wild' west!
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