We spent a day in Belfast. The big sight for us was the Titanic Museum. It's one of the best museums I've been to. I'm not really a fanatic for either Titanic (the movie), the historical period or massive iron ships but I enjoyed the museum.
I'm not really interested in "the troubles" so didn't stay longer than a half day in Belfast. After the museum, we went a short distance to Carrickfergus Castle. After maybe a 30-minute visit we were ready to move on. We met a local couple with a dog like ours where we had parked. After a long conversation we moved on. We stayed at a B and B overlooking the coast maybe two hours from Belfast.
We stayed a night in Derry/Londonderry. Again, the troubles aren't interesting to me, so we mostly explored the walls and area inside the walls.
Our favorite sights were coastal views, small fishing ports/villages. Eating fresh seafood, strolls/hikes along the coast, ancient history (stone age sights) and staying in B and B's was our basic itinerary. Northern Ireland doesn't have the influx of tourists that the Republic has. To me that was enjoyable.
Maybe my single favorite site in the entire area was Grianan of Aileach. It's a ring fort. We went in the morning from Derry/Londonderry and were the only one's there. Technically, we were a little before opening hours but there is no gate or ticket booth, just a walk up from a dirt parking lot. To me it was evocative; it made me think of a different era of Irish history. It was almost like going to Newgrange and having the place to yourselves.
We rented a car out of Dublin for $6 a day (yes, I thought there might be a mistake) and drove at our own pace. It was their smallest car, with a manual transmission. When the car was ready. The agent said in a voice loud enough for everyone in the lobby to hear, "Who has a small stick?" I thought that was funny.
The highlight for us in Northern Ireland was a day that included the Giant's Causeway, Bushmill's Distillery, Dunluce Castle and Royal Portrush Golf Course (about a week before they hosted the British Open). The four are close enough that the day didn't feel rushed. As always, we slept and had breakfast at a B and B a little up the coast then moved on to another B and B a little way down the coast.