Boris Johnson said he wanted to “change the country for the better” after he became the UK’s new prime minister.
Speaking outside Downing Street, he said the UK would leave the EU on 31 October “no ifs, no buts”, adding: “The buck stops with me.”
“The doubters, the doomsters, the gloomsters” who said it could not be done were “wrong”, the new PM added.
He also promised to sort out care for the elderly “once and for all”, and invest in transport and education.
Reforms to the social care sector have eluded previous governments because of their cost and complexity.
“We will fix it once and for all with a clear plan we have prepared to give every older person the dignity and security they deserve,” he insisted.
Mr Johnson also listed a wide range of domestic ambitions, including improving infrastructure and “levelling up” school spending, and reforms to ensure the £20bn in extra funding earmarked for the NHS “really gets to the front line”.
He pledged to boost the UK’s biotech and space science sectors, change the tax rules to provide incentives for investment, and do more to promote the welfare of animals.
The new PM will shortly begin announcing some senior cabinet appointments. It is believed he is about to meet ministers who he is planning to dismiss in his offices in Parliament.
Among those whose futures look uncertain include the Business Secretary Greg Clark and Northern Ireland Secretary Karen Bradley, both close allies of Mrs May.
There is also speculation about whether Jeremy Hunt will stay in cabinet amid reports Mr Johnson wants to move him from the Foreign Office but that his leadership rival had turned down the post of defence secretary.
Setting out his priorities for office, the former London mayor hit out at the “pessimists” who did not believe Brexit could be delivered and called for an end to three years of indecision.
“The people who bet against Britain are going to lose their shirts because we are going to restore trust in our democracy,” he said.
“The time has come to act, to take decisions and change this country for the better.”
He said he had “every confidence” the UK would leave the EU in 99 days time with a deal, but preparations for the “remote possibility” of a no-deal Brexit would be accelerated.
Mr Johnson vowed to bring all four nations of the United Kingdom – or what he described as the “awesome foursome” – together in the task of strengthening a post-Brexit country.
“Though I am today building a great team of men and women, I will take personal responsibility for the change I want to see,” he concluded.
“Never mind the backstop, the buck stops with me.”
The BBC’s Vicki Young said she was struck by the ambition of Mr Johnson’s objectives beyond Brexit and the fact that he would take personal responsibility for his success or failure in achieving them.
Labour’s Sir Keir Starmer said Mr Johnson’s speech was “all rhetoric” and the new PM needed to show leadership rather than the “glib” answers he had become known for.
The new Lib Dem leader, Jo Swinson, said she would welcome a cross-party push to find a solution on social care, but attacked Mr Johnson’s “bluster and bravado” over Brexit.
Mr Johnson took over after Theresa May handed in her resignation to the Queen. A number of her senior ministers have also already resigned, saying they could not serve under her successor.
Earlier, as she relinquished power after three years, Mrs May said being prime minister had been “the greatest honour” and wished her successor well.
In a farewell speech outside No 10, she said his government’s “successes will be our country’s successes”.
Mr Johnson’s audience with Queen Elizabeth II lasted more than half an hour.
During his journey to Buckingham Palace, his car was briefly held up by protesters from Greenpeace, who formed a human chain across The Mall.
Mr Johnson’s partner, Carrie Symonds, and key members of his staff were awaiting the new prime minister’s arrival in Downing Street.