If you want to go by public transport you could go by Bus Eireann or Goldline (Northern Irish) originating from Bus Aras bus station in Dublin 1 through a code share agreement. Getting the bus would be a cheaper and more lengthy trip than the train bus you get too see more of the countryside. They often offer good value day returns.
Across the road at Connolly Train Station you can catch a train to Belfast which will whizz you up far faster in about an hour so the mode of transport also depends whether you want to get their quickly or see more of the countryside as you go.
Alternatively you could go on a backpacker tour such as Paddywagons. However I've heard of one of their drivers coming on more than a bit fresh so if you're conservative it may not be your scene.
Or this crowd. Don't know anything about them while Paddywagon have been establised in the market for a long time & their buses look nice whatever about some of their drivers.
over 8 years ago