The best way to get around Ireland (unless you're on a tour or have hired a car) is by using Bus Eireann. It is a lot cheaper than travelling by train, and they go to more destinations around Ireland. From Dublin to Wexford (city) it takes approx. 2 hrs and 45mins (Although to somewhere like Gorey, in the north of co. Wexford, it would only take approx. 1hr and 50mins).