She unveiled a statue marking her visit, met jockeys and others involved in racing, and was shown stud horses at the centre in Kildare, west of Dublin.
Irish agriculture minister Simon Coveney said the Queen's trip had now entered a "more relaxing" phase.
She has expressed "deep sympathy" for victims of past British-Irish conflict.
The Queen is making the first trip to the country by a reigning UK monarch for 100 years and the first since it gained independence.
At a state dinner at Dublin Castle on Wednesday, she offered "sincere thoughts and deep sympathy" to people who had suffered during the two countries' "troubled past".
An apology had not been expected, BBC royal correspondent Nicholas Witchell said, but the Queen came "pretty close".
Continue reading the main storyThe Queen's among horses. She'll be happy. They don't bow, curtsey or fawn”End Quote Peter Hunt Royal correspondent, BBC News Earlier on Wednesday, her engagements had included a visit to Croke Park sports stadium in the capital, where 14 people were killed by British forces 91 years ago, and the earlier laying of a wreath at a service honouring Ireland's World War I dead.
BBC Ireland correspondent Mark Simpson said that having reached "half-time" in the Queen's four-day trip, there was now "no more politics - just horses and showbiz today".
The Queen is due to attend a variety show at Dublin's National Convention Centre on Thursday evening.
Mr Coveney said: "The first two days were very much about the emotive history of the two islands, reconciliation and moving forward and respecting the history. I think the Queen did that in an extraordinarily generous way.
"I think the second half of this visit is, I hope, more relaxing - and she's very interested in racing and horse breeding."
He said: "Horse racing is part of what it means to be Irish and an important industry for us here."
'Reconciliation'Addressing guests at the state dinner on Wednesday, the Queen opened her only public speech of the trip in Irish.
"A hUachtarain agus a chairde (President and friends)," she said.
The Queen opened her speech in IrishShe went on: "It is a sad and regrettable reality that through history our islands have experienced more than their fair share of heartache, turbulence and loss.
"These events have touched us all, many of us personally, and are a painful legacy. We can never forget those who have died or been injured, and their families.
"To all those who have suffered as a consequence of our troubled past I extend my sincere thoughts and deep sympathy."
She added: "With the benefit of historical hindsight we can all see things which we would wish had been done differently or not at all."
Continue reading the main storyI think people were very taken by the extent of her words and the clarity and ringing tone in which she said them”End Quote Enda Kenny Irish prime minister The Queen praised her host, President Mary McAleese, who had "done a great deal to promote this understanding and reconciliation" and "set out to build bridges".
Mrs McAleese, in her address to guests, said a new chapter had been opened between Ireland and Britain.
"The harsh facts cannot be altered nor loss nor grief erased but with time and generosity, interpretations and perspectives can soften and open up space for new accommodations," she said.
'Remarkable contribution'UK Prime Minister David Cameron was present at the state dinner, after earlier flying in and holding talks with Ireland's prime minister, Taoiseach Enda Kenny.
"As so often, her majesty has been able to put things and say things and do things in a way that massively strengthens the relationship," Mr Cameron said.
Mr Kenny said: "I think it was a remarkable contribution in what is a remarkable state visit.
"I think people were very taken by the extent of her words and the clarity and ringing tone in which she said them."
Sinn Fein president Gerry Adams, whose party did not attend the dinner, said he believed that the Queen's expression of sympathy was genuine.
But he added: "Queen Elizabeth's acknowledgement that the relationship between Britain and Ireland has not been entirely benign is a gross understatement.
"This will be forgiven if the future policy of her government is about building an entirely new future based on genuine equality, and mutual respect."
The Queen and Prince Philip's signatures on the visitors' book at the official residence of the Irish president
Wreath-laying at the Garden of Remembrance, the Queen (left) with President Mary McAleese (right)
<--The Queen inspects the Book of Kells at Trinity College
-->The Queen signs the visitors' book at Government Buildings, while meeting the Irish prime minister, on right
Irish veterans meet the Queen and the Irish president at the Irish War Memorial Gardens
The Queen is offered a pint at the Guinness Storehouse
The Queen and Prince Philip receive a hurley stick, at Croke Park stadium
The Queen wore a diamond harp brooch for her speech at Dublin Castle
The Queen views the stallions at the Irish National Stud
Thursday 19 May: Visit to National Stud at Kildare.
Friday 20 May: Tour of St. Patrick's Rock, Cashel and the English Market and Tyndall Institute, Cork.
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