2 adults and one teenager staying 3 nights in Dublin and two in Cork. What are the must sees? What is the cheapest way to get around? Should we try to get to England? This is probably a once in a lifetime chance for us. Best way to get from Cork to Belfast and then from Belfast to Dublin to leave?
Two weeks is barely enough time for Ireland, possibly you can cram in some Northern Ireland sights, but I would not attempt to get into England in the time available.
For what it's worth we spent two weeks in the southern half of Ireland in 2015 and another two weeks in north Ireland and Northern Ireland in 2016. I'd recommend you spend some time min the southern port cities and forego the North.