The National Library of Ireland is a great place to begin discovering your family history. They recommend that you read their information booklet "Family History Research: Sources at the NLI", then visit their Genealogy Advisory Service which offers free advice and resources for tracing your family. They have a number of searchable databases on computer and microfilm.
You will find a free guide to Tracing your Irish Ancestors at various locations, including Discover Ireland Centres /Tourist Information Centres. This has a lot of good tips on where to look in Dublin and online. The National Archives on Bishop St, Dublin and the Epic Emigration Museum's Irish Family History Centre also provide research help.