Mr Cameron will face his last Prime Minister's Questions in the Commons, before heading to Buckingham Palace to tender his resignation to the Queen.
He told the Daily Telegraph: "As I leave today, I hope that people will see a stronger country. It has been a privilege to serve the country I love."
After taking office, Mrs May will set about naming her own frontbench team.
The current home secretary, 59, was the only remaining candidate in the Conservative leadership contest following Andrea Leadsom's withdrawal on Monday.
The contest began when Mr Cameron, who has been prime minister since 2010, announced he would step down after losing the EU referendum in June.
Mr Cameron told the Telegraph: "I came into Downing Street to confront our problems as a country and lead people through difficult decisions so that together we could reach better times.
"As I leave today, I hope that people will see a stronger country, a thriving economy, and more chances to get on in life."
At midday Mr Cameron, who has said he plans to continue as MP for Witney in Oxfordshire, will face Prime Minister's Questions for the 182nd and final time as PM – his 319th in total as Tory leader.
Later, after the PM has tendered his resignation to the Queen, Mrs May will have her own visit to Buckingham Palace, when she will accept the monarch's offer to form a new government.
She will return to No 10 as the country's second female prime minister, following in the footsteps of Margaret Thatcher.