My best mate lives in a small town on the Antrim coast called Whitehead, just north of Carrickfergus (which is just north of Belfast), and have visited loads of times, both alone and with the family.
Belfast Zoo is pretty good (in north Belfast). Carrickfergus Castle is fun if your kids are into castles, knights etc. Ballycastle is pretty and has a few nice bars and restaurants. The Glens are good for a walk, and there's a country park in one of them. The drive north along the coast from Larne is great. Giants Causeway, obviously. There's the Gobbins and Carrick-a-Rede? rope bridge. Not been up to Portrush and Port Stewart.
One time we drove over to Donegal and around the Inishowen peninsular and Malin Head etc, which was great and is probably the easiest and nearest bit of the Republic.