Hello from Wisconsin,
Genealogy in Ireland. I believe the site to visit before you head off is 'rootsireland.ie'. I hired an Irish researcher, and he did a lot, found little but used rootsireland for much of his work. I don't know how far back in time you want to search, but records before 1810 are scarce, and other records were destroyed during the revolution.
For rootsireland You have to a pay to play, but a day's membership or a week's might be well worth the time and money so when you actually get to Cork (or where ever) you are doing more than basic research. I think you might find yourself paying to be on rootsireland while you are in Ireland. Each county of Ireland has a genealogy research center. You could contact the county you are interested in ahead of time.
As for visiting Cork and Belfast. They are at opposite ends of the country. Think five hours without breaks to get from one to the other. The really pretty area near Belfast is NW of Belfast away from Dublin, the Green Glens of Antrim and the Giant's Causeway. From Donegal head south to Connemara, a beautiful area of Ireland from Westport ( pretty place) to Leenaun then Clifden and ending at Outherard. from there head for Cork if your time is short.
wayne iNWI
If you go from Dublin to Belfast you could continue on around the top of the island getting into Donegal.