Dublin is the capital of Ireland. All Irish motorways begin here.
Hitching Out
Hitching out of Dublin is quite difficult. The easiest way is to take a bus to the outskirts in any direction and try to get a ride there. Buses are quite cheap.
If you arrived in Dublin (or Dun Laoghaire) by ferry, you should try to hitch a ride at the exit of the ferry port, as there are a lot of cars going to any part of the country.
Be sure to have a large card with the town name you want to go to spelt correctly and shown to oncoming drivers. If you need to stay in Dublin there's a free accommodation finder called Almara Accommodations Dublin.