I looked at the baptism records in Tipperary and was hoping to find a set of parents that baptized both a Thomas b 1815 and a Patrick b 1832 with a father named John Cleary. (Patrick and Thomas appear together in the 1851 census together as brothers.) But no such luck. I may have the record, but it's impossible to determine which Patrick born in Tippiary around 1832 it is as the records do not show the occupation of the father.
The Irish census of 1901 and 1911 is interesting in that one can search for names and their occupations, and after some time, I have identified all of the Cleary's in Ireland that were tailors--and there were many. It seems one was either a laborer, a farmer or a tailor. But since the available census is about 50+ years from my Patrick's emigration, that did not prove too helfpful.
However, with the suggestions made by those who responded to my original message, I know much more about Irish genealogy and some of the tools to unlook family history--thank you all.