Stephen,
I found the births on the Irish civil indexes site, which is in the Familysearch collection. You can access it from the US too:
https://familysearch.org/search/collection/1408347
Your problem is that the registration of births and deaths (and RC marriages) in the statutory civil registers only started on 1.1.1864. So you won?t find any of the children who were born before that, on that site. (And that?s why I only gave you the details of Henry and Eliza Jane).
When you get those baptisms, let me know the townland (ie address), and the father?s occupation, and I?ll see if I can help find where they lived, by using Griffiths Valuation.
For births prior to 1864, you have to rely on church records where they still exist. (Lots of Irish church records have been lost over the years for a whole variety of reasons, fire, negligence, water damage etc). I would expect your family?s baptisms to be in the Ballyscullion RC records. Their baptisms and their marriages start in 1843 so that looks promising. They appear to be on-line on the rootsireland site, so you can search there. (I say appear to be because their index doesn?t refer to Ballyscullion parish at all, but they do have Bellaghy (which is not the correct name) but I assume they must be the right records all the same).
http://ifhf.brsgenealogy.com/index.php
There is also a set of the parish records in PRONI (the public record office) in Belfast, which you might find interesting if you want to see the original handwriting in the parish register (as opposed to Rootsireland?s information which is just a typed transcript).
Don?t expect too much information from the RC baptism records for the mid 1800s. All you generally get is the child?s name, the parents names and the two sponsors. No addresses, occupations or anything like that. Some are also in Latin. Likewise with RC marriage registers, all you get are the couples names and their two witnesses. So no parents names, addresses or other personal information.
You ask where to focus your efforts when you visit. As far as records are concerned, the majority of official records are kept in PRONI, the public record office, in Belfast. A lot of the normal genealogical records eg censuses, have been destroyed and so you have to rely on things like school attendance records and so on. But to try to trace them we need to know where exactly the family lived, which is why I need the townland from the birth certificate(s). The father?s occupation also matters. If he was an agricultural labourer/weaver (as I suspect) there may not be much, but if he was a farmer we might be able to find out a bit more. He might even have left a will.
I would think you might want to visit the church the family attended and married in, and a look at gravestones in the relevant graveyard(s) would also be a good idea.
Elwyn