Im so sorry Kiba, I completely missed this topic (I rarely come into GD these days, which is completely ignorant on my part!).
Any way I can help make your trip more awesome I will try! Frist off, I really hope you all have a good time! Ireland is an amazing country, with beautiful scenery, and has its own unique history. I will cut straight to the point with my own thoughts and opinions, but I will go on the assumption that you want to enjoy Ireland for what it is known for; green fields, castle ruins, historical places, pints of Guinness, a peat fire and bowls of stew! A bit stereotypical yes, but that's what makes this country so great, and experiences which I really don't take full advantage of.
With that in mind, I would cross 2 places off of your list to spend much time in, and that's Belfast and Dublin. First off, they are major cities, and not different to any other major city in Europe. Sure, you will have nice architecture and maybe a museum and stuff, but what city doesn't. Unfortunately, in terms of national identity, Dublin has been completely inflused with migration, and as the majority of the country jokes, you will be hard pressed to find an Irish person in Dublin anymore. I don't mean that in a discriminatory way, its just the way it is.
Belfast, and the North.... I don't know if I need to explain much about up here in these parts, but the history that exists in the North is always on full view, and tours across various interfaces are available. I live here, so our past is not appealing to me at all.
In terms of where I would go in Ireland, if I had a choice is very easy - all of the West coast. Starting at the top we have Donegal, which is one of the most beautiful places on this planet. I want to retire out there. its just so beautiful. Then moving down to Galway, Sligo, County Clare..... just so many beautiful small towns and proper salt of the Earth people, and so many historical sites to see along the way. Even the Father Ted house if that's on the list! I will admit, I have never been to Cork in my life, which is a bit weird, but at the same time, I cant say that an opportunity has ever presented itself to go, nor have I ever had any desire to go.