Basing yourself in Belfast is probably the best way to go. Not only is the capital easily accessible via its two international airports, but it also offers museums and fine food in the Titanic Quarter, the Ulster Museum, and the restful Botanic Gardens. But don't stay too long, because there's a lot of ground to cover.
Driving north along the Carrickfergus road, you'll reach Larne in just over an hour. It is worth leaving an hour or two to visit Carrickfergus Castle. After Larne, make a detour to see the gorgeous Glenou Falls, then strike out north past Ballymena into Antrim to make a date with a giant.
Stop off at the Giant's Causeway to see the remarkable basalt columns (and hear the legends attached to this mythical spot). Then head west along the coast via Bushmills (a whiskey tour is optional, but recommended), and try not to be detained too much by the beauty of the coast. If you need to take some Insta-ready snaps, Mussenden Temple is the place to head, with its 18th-century chapel and stunning cliff-top location.
Now it's time to head inland. Leave some time to visit Derry, with St. Columb's Cathedral and museums telling the city's often bloody story in fascinating detail. Then keep on for the interior, passing into County Fermanagh where Lough Erne takes centre stage. It's a wonderful spot for country walks, especially around the Cliffs of Magho.
After Lough Erne, it's time to head east through rolling countryside and hills until you hit Armagh, where you can stargaze in the excellent planetarium. You can keep going into southeastern Northern Ireland to meet the beautiful Mountains of Mourne, an outdoor wonderland crammed with dramatic lakes (and amazing golf courses if that's your passion). After that, completing the circuit is easy, with a quick run up the A24 to Belfast.