Lots of levels of walks, Hen Mountain is a nice one to start and kid friendly, it's right in Hilltown. Slieve Donard is the largest and the largest in Ulster, 7th highest in Ireland at 850metres (2,789 feet).
Be sure to check out the 'top' car park, and the Cloughmór stone. Then wander around the side of the mountain to Kodak corner for a breathtaking view of Carlingford lough.
Nice large park in Newcastle and starting point for climbing Slieve Donard