We just did exactly this in May.
We spent 3 nights in Belfast and 4 nights in Dublin. Yes, I know Belfast isn't technically part of Ireland, but....
We flew into Dublin and took the bus from Dublin airport to Belfast. (It's only a couple of hours.) We had the afternoon and evening in Belfast, and we just walked around and explored. The next day, we did a tour to the Giant's Causeway and other sights on and near the Causeway Coast. It was gorgeous. We did a private tour with Glenara Elite Travel for this. The day after, we walked around and visited a couple of churches and such. We would have liked to have visited City Hall, but it was closed that day. In the late afternoon, we did a murals/troubles tour with Paddy Campbell (Black Taxi) tours. I highly, highly recommend this if you go to Belfast.
We took the bus to Dublin the next morning. Train would have worked, too, but the bus station was just around the corner from our hotel in Belfast.
That afternoon and evening, we walked around and explored some of Dublin. The next day, we did a tour to the Boyne Valley, Trim Castle, Fore Abbey, and Loughcrew tombs. We really liked that. The day after, we did a tour to the Rock of Cashel, Blarney Castle, and Cahill Castle. It was excellent, as well. Both were with Irish Day Tours. The last day, we toured Kilmainham Gaol, St. Patrick's Cathedral, the Book of Kells, and Christ Church Cathedral.
If you don't want to go to the North of Ireland, then I would suggest choosing one other city for a base, such as Galway or Cork, in addition to Dublin, and book some tours from those places.
You can get to the Rock of Cashel, Blarney Castle, etc. by public transport, but it is a bit more complicated and time consuming. This is why we decided to do day tours. They were long days, but very good.
Be aware that almost everything in Ireland needs to be booked in advance, even restaurants, even on weeknights. Kilmainham Gaol, the Book of Kells, etc. usually need to be booked at least a day in advance (the gaol WELL in advance), or you will be out of luck.
Hope that helps!