If I know only that my ancestor was born in Ontario, how can I possibly find out what district the family was in? (This is a family that first went from Ireland to Canada then later moved on to the United States).

See OntarioGenWeb's Beginner's Guide for tips on "where to start" when you don't know where in Ontario you should be looking.
[From Ang] -- Have you checked through the LDS's Canadian Index for their birth? You can get this through interloan or through your local LDS family history center. Depending on the year you can look up births from as early as 1869. Another good place to look is the online 1871 canadian census. Hope this helps.