Mari, we took the train from Dublin Connolly Station to Belfast Central, no reservations just walked up. It's about 10-12 euro per person each way. It takes a little over two hours for the trip, and I arranged for a Black Taxi Cab driver to meet us at the station, and we took a tour of some of the Troubles areas for a couple of hours. It's a great way to get a bit of history and perspective from a local, and was not very expensive, I think in the 70-80 euro range. We asked John to drop us near Belfast City Hall where there are free tours once or twice daily-it's a great building and worth seeing.
There is good public transport in Belfast, too.