Notes:

There are two separate mainline stations in Dublin that connect to the north and south of the island respectively. There is no direct rail connection between the stations but there is a separate tram line (LUAS) that runs between the stations. Passengers must exit the station to access the tram line. 

In the Dublin region there are three separate and distinct rail networks.
1. Main line rail (handled by Irish Rail or Iarnróid Éireann)  connects to the outer suburbs, for example to Maynooth and Sallins, see Dublin Map.
2. DART (Dublin Area Rapid Transport) service which is a commuter line running along the east coast 
3. LUAS (Light Rail) there are RED and GREEN lines that extend to the south and west of the city.

See link for carrying bicycles on trains. It is important to book bikes on the trains especially trains not running to Cork and Belfast. On other lines there is limited capacity for bikes. Book in advance and arrive at the station early. 
NB: On Dublin Commuter Lines there are time restrictions on when bikes can be carried.
NB: Bikes are not allowed on the LUAS tram lines at any time of day.
NB: Bikes cannot be carried on buses, either dublin city buses (run by Dublin Bus) or intercity buses (run  by Bus Éireann).

Irish Rail staff have told us that passengers sometimes arrive with bikes and without booking their bikes in advance. They are generally not turned away but can take up the spaces that others have pre-booked. So book in advance and arrive early. It is safer to lock the bikes on the train and select seats near your bikes.
Note that bikes cannot be carried on city and suburban buses.