Fill out the forms, available through the Department of Foreign Affairs web site.
Supply the required identification. There are several different documents required.
Attend an Interview at one of the Departments offices or at an Embassy in the country you are in.
You don't need a passport to travel between the UK and the Republic of Ireland.
unfortunately you do
yes there are no border restrictions from the republic,
Passport
No
passport or driving license if you are from the UK
yes, because the republic of ireland is a seperate country from the UK you will need a passport to go there.
Generally you will need a passport to visit Ireland. As a general rule, you should always get a passport when travelling internationally. The fact that you grandmother is Irish, does not mean you can travel to Ireland without any passport. You may actually be entitled to get an Irish passport. See the related question below for details of documents needed to visit Ireland.
No. They are both in Ireland, so if you are in Cork legally, then going to Dublin does not require any additional documentation. You may have needed a passport to get to Ireland in the first place. If so, then keep it with you when travelling around Ireland.
depends where your coming from, see the related link.
No, the United Kingdom has not signed up to the Schengen agreement. A passport (or visa, if needed) is required to enter the UK when travelling from a Schengen country. One anomaly to this is the land border between Northern Ireland and the Republic of Ireland (i.e. the international border between the UK and Ireland). This is effectively an open border to vehicle traffic without passport checks, even though the Republic of Ireland is a Schengen state.
the republic of ireland, its of one not in one