Joyce, you can serarch the parish registers online from your home. The link is http://registers.nli.ie and each page (or pair of pages) in the parish register comes up as a downloadable photo file, so you can download a whole seciton of the register and search at your leisure. There's also an interactive map which lets you see what other parishes are near the one you're searching, in case you want to search them. It can be a bit tedious to search a large number of pages, but it's a massive improvement over going to the parish (or the library) to view the records..
I don't know whether my gg-grandfather ever emigrated, but I suspect not, because his children and grandchildren (including my grandmother) were all born in Mayo. However, I do know from a history book I have which was written about eastern County Mayo that many familes went annually to England or Wales to find seasonal work, and some did stay there, and I suspect that was also true for neoghboring parts of Roscommon. My mother's father was born in Wales while his family (also from Mayo) was there, though they went back to Ireland, where he grew up. The siblings of my Mayo grandmother mostly came to the US (to New York, and many descendants are still there), but two went to England. One went back later to farm the family land in Mayo, after the economic situation improved, and they are still there. The other stayed in England and married a man named Broderick, but I've lost track of them (I've never met them, although I've visited the ones in Mayo).
If you want to look at my family tree at the FamilySearch site, that's where I have the most info. It sounds as though you're somewhat familiar with the site. The Patrick McDonnell mentioned above has ID number LRQV-3BB at the site, and from there you can go up and down the tree (you won't be able to see me, because they don't allow one to view the records of living people unless you're the one who input them). I'm not a Mormon myself (it's their site), but I use the site because it is the largest one in the world, one can search for amazing kinds of info there, and the records are arranged in a way that's really easy to use, and of course it's free!
I had my DNA tested at FamilyTree DNA (no relation to the FamilySearch site). They say that they keep the info confidential, unless one chooses to share it with others, but I suppose one never knowws. I'm in my 60's, so at this point I don't really care who has my DNA, and I've uploaded my data at a number of sites, because I'm still trying to find the birhplaces of two of my great-grandparents.
Take care and feel free to keep in touch if I can be off help. My direct email is kevin45sf@yahoo.com