The Mournes are the kind of mountains that get under your skin. Granite peaks, sweeping valleys, and a dry-stone wall called the Mourne Wall that runs 35 kilometres across the summits, built between 1904 and 1922 to enclose the Silent Valley reservoir catchment. The scale of it is hard to believe until you're up there looking at it. Slieve Donard at 850 metres is the highest peak in Northern Ireland and accessible to reasonably fit walkers in about 4 hours return.
Start from Newcastle town and follow the River Glen south into Donard Park and up the mountain. The path is well-worn but steep in the upper section. The views from the summit cairn on a clear day genuinely include Scotland to the east, Wales to the south, and the Isle of Man in the Irish Sea. Check the weather before you go. The mountains create their own conditions quickly and the summit can be in cloud when the town is sunny. There are parking charges at the main Newcastle access point.