Irish emigration before An Górta Mór [The Great Hunger] was mainly, though not exclusively by the more ‘well to do’, mostly non-catholic sector of the population.
There are very few records with identifiable detail of this traffic and it’s mostly at ‘port of entry’. Little regulation or information collection requirements existed for port of departure at this time and most of the information which was taken is lost.
One such collection can be found in Irish Emigration Lists, 1833-1839: Lists of Emigrants Extracted from the Ordnance Survey Memoirs for Counties Londonderry & Antrim
Compiled under the direction of Brian Mitchell, extracts of which are viewable on Google Books
Another series of books of early emigrants is Erin’s Sons: Irish Arrivals in Atlantic Canada, 1751-1858 ~ Terrence M. Punch [4 volumes], some extracts on Google Books
There are some earlier lists are on Irish Passenger Lists, 1803-1806: Lists of Passengers Sailing from Ireland to America : Extracted from the Hardwicke Papers ~ Brian Mitchell and finally some later lists Irish Passenger Lists, 1847-1871: Lists of Passengers Sailing from Londonderry to America on Ships of the J. & J. Cooke Line and the McCorkell Line ~ Brian Mitchell
Further reading: Finding your Irish Ancestors ~ Brian Mitchell