Best place and best schools for UK / Irish in NY (New York)

Hi. We are from Northern Ireland and considering potential options to move to NYC. We have 2 boys aged 6 and 8. My husband would work in mid-town Manhattan. We are considering moving to Manhattan or Brooklyn. We really do not know NYC but feel that Brooklyn may be more chilled out, a bit cheaper and better for the family? Should we consider Queens? We then also need to think of schools for the boys. We were thinking of St Bernadette in Brooklyn? Or Lady of Pompeii in Bleecker Street? Do we need to apply to a lot of schools to get in? Is there big difference in quality between the schools? Is it possible to avoid the high private school fees and still get into public school. if we move we would arrive in Sept or Jan? Realize quite a big question and no easy answer but much appreciate any inputs! Claire