The Taoiseach is to bring a memo to Cabinet tomorrow on the UK's decision to leave the European Union.
Enda Kenny has identified travel and trade as being the main issues of concern to Ireland as a result of the Brexit vote.
Speaking in Claremorris in Co Mayo, he said there was now clarity as to when the British government would trigger Article 50 of the Lisbon Treaty to leave the EU.
Mr Kenny said this meant that Ireland had to be clearly focused on the impact of any negotiations, to ensure its best interests were served.
He said that the vital priorities centred on common travel and trade issues but that there were a host of other matters that had to be taken into account.
Circumstances relating to the peace process and the situation regarding the border were also a key consideration, he said.
The Taoiseach said the Government had been preparing for Brexit for quite some time and was taking into account a range of opinions from several different sectors.
Describing the UK withdrawal from the EU as one of the "greatest challenges to face Europe and Ireland for many years", Mr Kenny said he wanted to ensure Ireland maintained very good relationships with Britain but that it is also to be central to the manner in which the EU will move forward in the years ahead.
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It is likely to take two years for Britain to completely disengage from the EU.
UK finance minister Philip Hammond this morning said that the UK economy will face turbulence as the government negotiates the country's exit from the EU.
He said he was in favour of getting a deal done "as soon as possible".
Separately, Tánaiste Frances Fitzgerald said there are challenging times ahead but nothing has changed in terms of the goals of the Irish priorities.
She said it is important to protect Ireland's trade.
Ms Fitzgerald said that the British prime minister laid out her approach to the negotiations at the weekend and Ireland will negotiate strongly to protect its position both with the UK and the broader membership of the UK.
Earlier, Northern Ireland Deputy First Minister Martin McGuinness described as "disturbing" the Brexit plans outlined by Ms May yesterday.
Mr McGuinness said the referendum result showed that the United Kingdom is anything but united, given that Northern Ireland and Scotland see their future in Europe.
Speaking to RTÉ's Morning Ireland, Mr McGuinness said: "It's quite clear that whenever the negotiations begin, if they do begin in March of next year, that the British government are faced with a very tough negotiation.
"So I think we're headed to a head-on collision between the British government negotiators and European Union."
He added that Sinn Féin is totally opposed to what the British government is trying to do.
Mr McGuinness said the negotiations between the EU and UK had not yet begun, which places Northern Ireland and Scotland at a serious disadvantage.
He added that there is very little sympathy for Britain in Europe and staying in the single market would be difficult.
Mr McGuinness said that Sinn Féin would not give in on the issue and that the Irish Government would have a significant role to play in negotiations next year.
"I think the Irish Government have an important role to play in standing up for the democratically expressed wishes of the people of the North, who wish to remain."
Ireland facing greatest foreign policy challenge in years - Flanagan
Minister for Foreign Affairs Charlie Flanagan has said Brexit is the greatest foreign policy challenge to Ireland since it joined the European Economic Area 43 years ago.
Mr Flanagan said that he welcomed the clarity Ms May has given the situation by naming a date for triggering Article 50.
"Our priorities here are quite clear - the economy, the fact that we have in excess of €1.2bn trade with the UK every week, obviously the situation regarding Northern Ireland.
"The preservation and maintenance of the Common Travel Area which we have enjoyed with the UK since the foundation of our State. And of course the fact that we still remain active members of the European Union."
Democratic Unionist Party Chief Whip Jeffrey Donaldson said that Brexit could be a good thing for Northern Ireland and the party will work to achieve that.
Speaking on RTÉ's Today with Sean O'Rourke, Mr Donaldson said that while Brexit brings challenges, the DUP will work to ensure Northern Ireland is treated as a special case.
Mr Donaldson said that the DUP did not see the need for another North-South body to deal with the issue and an All-Ireland forum would undermine the North South Ministerial Council.
He added that the Taoiseach is "not the prime minister of Northern Ireland" and that Mr Kenny should not have proposed an All-Ireland forum without more dialogue with parties in the North.
The chief executive of the Irish Exporters Association has said it is absolutely crucial that Irish exporters start to do scenario planning and risk assessments.
Speaking on RTÉ' News At One, Simon McKeever said: "With the economy in the UK looking like it's going to slow down, the risk is definitely to the upside.
"The currency is the single biggest thing that is worrying [exporters] at the moment."
Mr McKeever also said the Government should address the issue of a hard border urgently.