Here are just a handful of suggestions:

In and around Newcastle; the Mourne mountains, Murlough beach, Tollymore Forest Park, Dundrum Castle, Castlewellan Forest Park.

Belfast: the peace wall, murals, museums, bars, black cab tour, Crumlin Road Gaol, Titanic Museum.

Derry: Free Derry corner, city walls, museums and bars.

One hour from Belfast (or Derry): Dark Hedges, Giants Causeway, Carrick-a-rede rope bridge.