Cris,
It probably depends on exactly when the family left Leitrim. The railway from Dublin reached Sligo in 1863 and Bundoran (from Dundalk) in 1866. If they lived near either of those places and left after the railway was operating I’d guess they took the train to Dublin. Then from there, there were regular sailings to Liverpool or Heysham. They would then get a train to Hull.
According a 1922 train timetable, at that time, Sligo to Dublin took around 5 hours. Liverpool to Hull was around 4 to 5 hours depending on the timing of the connections (you had to change at York I think).
If the family left before the train had arrived in their area, I’d say they may have got a coach to Dublin. Or if they were not very well off, they may have sent their luggage ahead with a carrier and just walked. Many did that. There were also some direct sailings from Sligo to Liverpool which is another option. (In 1848 the SS Londonderry was on that route and 150 perished):
https://warwick.ac.uk/fac/arts/history/chm/outreach/migration/backgroun…
The most likely port of departure for New York is probably Liverpool. There were sailings from other places but Liverpool acted as a sort of clearing house for migrants from all over Europe and frequently had daily departures to America. Sailing ships were rapidly disappearing by the 1870s and most were steam which took about 7 days.
