The Pope will lead cardinals in his first Mass, begin appointing senior Vatican staff and may visit his predecessor, Benedict XVI, Pope Emeritus.
Pope Francis I is the first Latin American and Jesuit pope and has received a flood of goodwill messages from around the world.
He also faces a series of tough challenges.
The Church has been dogged by infighting and scandals over clerical sex abuse and alleged corruption.
Thursday morning saw Pope Francis I beginning the day with a visit to Rome’s main basilica dedicated to the Virgin Mary, Santa Maria Maggiore, for a private prayer.
“He spoke to us cordially like a father,” Reuters news agency quoted Father Ludovico Melo, a priest who prayed with the Pope, as saying.
“We were given 10 minutes’ advance notice that the Pope was coming.”
The election of Cardinal Jorge Mario Bergoglio, Archbishop of Buenos Aires shocked many onlookers when it was revealed on Wednesday.
Although he reportedly came second to Pope Benedict XVI during the 2005 conclave, few had predicted the election of the first pope from outside Europe in 1,300 years.
Pope Francis I will return to the Sistine Chapel on Thursday afternoon, scene of his election, to celebrate Mass with the cardinals.
Over the weekend, he will meet the world’s media at a special papal audience, an opportunity perhaps to set out some of his global vision.
Pope Francis I had been greeted by crowds roaring their approval when he appeared at the balcony overlooking St Peter’s Square on Wednesday evening, about an hour after white smoke rose from the chimney of the Sistine Chapel to announce to the world that a new pontiff had been elected.
“It seems that my brother cardinals have gone to the ends of the earth [to find a pope],” Francis said wryly, referring to his native Argentina.
“Now, we take up this journey… A journey of fraternity, of love, of trust among us,” he said.
Pope Francis I endeared himself to the crowds – and underlined his reputation for humility – when he asked them to bless him before blessing them in return.
Later, according to the New York Archbishop Cardinal Timothy Dolan, Pope Francis I shunned a special car and security detail provided to take him to the Vatican – “I’ll just go with the guys [cardinals] on the bus,” Cardinal Dolan quoted him as saying.
At the dinner itself, Cardinal Timothy Dolan said the Pope had made the cardinals laugh when he referred to the seven days of meetings that led to his election, saying: “I am going to sleep well tonight and something tells me you are too.”
The 76-year-old from Buenos Aires is the first pope to take the name of Francis – reminiscent of Francis of Assisi, the 13th Century Italian reformer and patron saint of animals, who lived in poverty.
He faces a gruelling schedule over coming days, with a visit to his predecessor Benedict XVI at his retreat at Castel Gandolfo outside Rome reportedly planned, as well as audiences with his cardinals, the media and the faithful.
The visit to Benedict is important, correspondents say, as the existence of a living retired pope has prompted fears of a possible rival power.
The Pope will be installed officially in an inauguration Mass on Tuesday, March 19, the Vatican said.
His election was met with thunderous applause at the cathedral in Buenos Aires and with delight and surprise elsewhere in Latin America – home to 40% of the world’s 1.2 billion Catholics.
Guillermo Lopez Mirau from Salta, Argentina, said he was delighted with Cardinal Jorge Mario Bergoglio’s election.
“People here are overjoyed. You can hear sirens and church bells ringing in the air.”
President Barack Obama sent “warm wishes” on behalf of the American people to the newly elected pontiff, hailing the Argentine as “the first pope from the Americas”.
The new leader of the world’s Anglicans, Archbishop of Canterbury Justin Welby, said he was looking forward to “walking and working together”.
And Argentine President Cristina Fernandez de Kirchner – who is said to have clashed with the Argentine archbishop in the past over issues including gay marriage – wished the pontiff a “fruitful pastoral mission”.
Pope Francis I takes the helm at a difficult time for the Catholic Church, facing an array of challenges which include the role of women, interfaith tensions and dwindling congregations in some parts of the world.
Cardinal Jorge Mario Bergoglio, who was not among the frontrunners before the election, is regarded as a doctrinal conservative.
But he is also seen as a potential force for reform of the Vatican bureaucracy, which may have won the support of reforming cardinals.
Pope Francis I will come under strong pressure to reform the Curia, the governing body of the Church.
Pope Francis I’s next schedule:
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