Here are some links to some significant research resources for Irish genealogy. While many records have been digitised they have not always been transcribed accurately so to be thorough you should always check the original source and also visit local archives, family history centres, graveyards, libraries etc. Search for typos and other spelling variations of the names you are looking for. It is also helpful sometimes to transpose the first and family names - particularly in census and government records.
There is much value in examining records in context rather than looking at filtered search results because it will help you understand the wider picture but also might lead to breakthroughs via serendipity or observations of broader records. For example, sometimes relatives might get married on the same day - you may not notice this if only looking for one record.
You should be aware that you cannot do comprehensive genealogical research purely over the Internet and certainly not without spending any money. That said, you can start your journey with free online resources.
I would suggest starting with;
Irish Genealogy - Free church & civil records provided by an Irish Government project
Irish Census - The 1901/1911 census records, with the few surviving records
Lennon Wylie Site - great for NI / Belfast centred research. Mary Lennon is a genealogical hero.
Ancestry Ireland (The Ulster Historical Foundation)
Emerald Ancestors (BMD Data)
Roots Ireland (BMD Data from IFHF)
General Register Office NI (GRONI)
LDS Pilot Project - Irish Civil Registration Indices 1845-1958
Irish Genealogy (Free church & civil records provided by an Irish Government project)
Public Record Office NI (PRONI)
Belfast Family & Community History (1901 & 1911 partial census)
National Archives (Kew)
My Roots, Tracing your Belfast Ancestors (Free PDF booklet)
A History of Ulster, Johnathon Bardon, Blackstaff Press; 2nd Ed. (2005)
The Irish in Britain, 1750-1922 (University of Bradford, Irish Diaspora Studies)
Family Search (from the LDS)
The Plantation of Ulster (BBC)
Mary Lennon and Ros Davies are deserving of high praise for their genealogical resources.