Below you'll find a list of Irish Government agencies that you may come in contact with, or need the services of, before you move to Ireland. While living in Ireland you will certainly find yourself in contact with many of them soon after your move.

Revenue (Revenue.ie)
Revenue is the name by which the Irish Tax and Customs agency is known. They handle all tax related affairs for business and personal purposes, as well as customs enforcement.

Welfare (Welfare.ie)
Welfare are responsible for social welfare services in Ireland. The website contains information on benefits available to all residents of Ireland. You will also find jobseeker and employer information, and they have a whole page dedicated to people moving to and from Ireland. This is especially useful for people moving to Ireland who need to know if they are entitled to live here and what they can do when they arrive.

Citizens Information (citizensinformation.ie)
A website providing information on public services and entitlements in Ireland. A wide array of topics are covered including education, housing, money, health, justice and more. They too have a section for movers which covers Visa requirements, becoming an Irish citizen, importing a car to Ireland and other topics.

Department of Foreign Affairs & Trade (DFA.ie)
Passports, citizenship, consular advice, business opportunities and more.

Irish Naturalisation & Immigration Service (inis.gov.ie)
More information on immigration, naturalisation, asylum, repatriation, investor schemes, etc.

Government of Ireland (gov.ie) - self explanatory.

Dept. of Agriculture (agriculture.gov.ie)
Bringing a pet with you when you move to Ireland? You will need to check out this link.

There are more, many of which you will find at the link above for Citizens Information. I have found that it can be confusing at times when dealing with Irish government agencies. For example, many of them give information on the same topic, and sometimes it can be contradictory. It can be frustrating trying to find the source of truth, and this is especially true when a large part of your move depends on the information you receive (e.g. visa's, pet travel info etc).

Please add any relevant information about this topic in the comments, or feel free to open a new Group.