A large number of Northern Ireland records are available through PRONI http://www.proni.gov.uk/, but each record can be a different story.

You might get some ideas from my chat with Megan Smolenyak http://irishfireside.com/2010/12/17/irish-genealogy/

If you visit PRONI the Premier Inn Hotel Titanic Quarter is within walking distance. Here are a few other resources:

http://www.genuki.org.uk/

http://www.irishroots.net/

http://www.cyndislist.com/uk/irl

https://www.emeraldancestors.com/databases/

Moy is in Co. Tyrone: 

http://www.cotyroneireland.com/

http://tyrone.rootsireland.ie/

Try to do as much "homework" as you can before going to Ireland.