First question is, where they really 34 years difference in age? 1862 & 1828?www.theshipslist.com
has passenger records but ancestry.com has the biggest collection.
There were 132 Tracy's in Galway in 1855, variety of parish's, do you have Patrick and Bridget's 2nd. son's name?
Usually this would be the name of Bridget's father!
There were 78 Flannery's in Ireland, 78 in Galway.
Both were born before 1864 the time of civil registration so any possible record of baptism would be a parish record.
The Irish History Foundation could be accessed with as little information as you have, but I'm sure in the case of Patrick Flannery you may get multiple records. www.rootsireland.ie
is a pay website but you will find that is true of most irish genealogy.
These resources won't be of much help to you but will show you what is available for research.
Tithe Applotments: 1823-1837 (this was for taxes) www.genealogy.nationalarchives.ie
Griffith Valuation (of land) 1847-1864 (for taxes)www.askaboutireland.ie
and a simpler form onwww.failteromhat.com
1901/1911 Census: www.census.nationalarchives.ie
Only bits and pieces of census prior to 1901 exist. There was a fire in 1922 that destroyed the census and some parish records.
Note in the current telephone book for Galway are 41
Flannery's and 7 Tracy's. Too many Flannery's without some idea of where Patrick came from. Tracy, send 7 letters with
the details you have and perhaps you might find someone who
can give you a hand.