In Dublin the major repositories of records are:
Office of the Registrar-General
In Belfast the major repository is:
Registry of Deeds in Dublin, and
Mention should be made of branches of the Family History Library (LDS) in Dublin, Belfast, Cork, and Derry.
In a few years you will be able to let your finger do the walking via the Internet.
Census Records | Vital Records | Family Trees & Communities | Immigration Records | Military Records Directories & Member Lists | Family & Local Histories | Newspapers & Periodicals | Court, Land & Probate | Finding Aids