"Britain needs to show the world that we can regain our stability once again," Burnham said in a speech outside 10 Downing Street on Monday after succeeding Keir Starmer as British leader.
"We have not been good enough, and we need to be better."
Promising Britons "some breathing space now", he said the government would set out some measures to tackle the high cost of living, and how it plans to fund them.
Further out, he said: "We will help more young people into work by changing the education system and giving them more support, more mental health support, and we will build more council homes.
"That is the fair and sustainable way to bring the welfare bill down, to meet our fiscal rules, and to honour our commitments on defence to our international partners."
After weeks as leader-in-waiting, Burnham officially became prime minister during a meeting with King Charles III at Buckingham Palace, where the monarch asked him to form a government following Starmer's resignation.
Buckingham Palace released a photo of the king and Burnham shaking hands.
The ceremony known as "kissing hands" marks the transfer of power.
Burnham is the seventh UK prime minister since 2016 in a now well-rehearsed handover after replacing Starmer as leader of the centre-left governing Labour Party on Friday.
Starmer earlier made his farewell speech outside 10 Downing Street before resigning, telling supporters that "my work is done" and that he was leaving with "good grace" and a "smile".
with AP
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