But Jeremy Corbyn, leader of the country's main opposition Labour Party, said May had read out a long list of problems, but failed to provide any solutions.
May set out her own vision for a trading relationship Britain after it leaves the European Union, spelling out five tests, and insisting membership of both the European Single Market and Customs Union would come to an end.
"We are leaving the single market. Life is going to be different. In certain ways, our access to each other's markets will be less than it is now," said May.
She also ruled out a hard border between Northern Ireland and the adjoining Irish Republic which will remain a member of the European Union (EU).
Reaction to May's speech was mixed, but it was broadly welcomed with optimism in some quarters.
Carolyn Fairbairn, director-general of the CBI, said May's speech rightly set the bar high for the economy and jobs.
"We heard the strongest acknowledgment yet of what's needed to get a good deal," said Fairbairn, adding membership of some EU agencies, willingness to take steps to guarantee a level playing field were welcome softening of red lines.
"They will make a good deal more achievable," said Fairbairn, "But more is needed to lift the fog of uncertainty and we welcome the PM's call to 'get on with it'."
Catherine McGuinness, policy chairman at the City of London Corporation, also welcomed May's speech saying it is a step towards an ambitious free trade agreement.
"The UK shares the same regulatory starting point as the EU, an advantage that no other potential trading partner has. And so the Prime Minister is right: the UK should look to secure a bespoke deal with the EU.
"We look forward to securing a transition deal with the EU as soon as possible and then discussing these trade proposals further. Securing this type of deal would see Europe retain access to London -- one of the world's leading financial hubs."
Labour's Corbyn said: "After 20 wasted months, Theresa May has once again failed to bring real clarity to the negotiations -- and worryingly, she admitted that her approach will reduce our access to European markets. She read out a long list of problems but failed to provide solutions, particularly on the urgent question of preventing a hard border in Northern Ireland. Theresa May's only clear priority today seemed to be to tie the UK permanently to EU rules which are used to drive privatisation and block support for British industry."
Vince Cable, leader of the minority Liberal Democrats criticised the speech, saying May avoided tough decisions.
Cable said: "Theresa May has once again prevaricated from making serious decisions about our future. Her speech outlined all the reasons why we should stay in the single market and customs union."
There was no early detailed response from the EU, but Pieter Cleppe, Brussels head of Open Europe think tank, said May's speech provided some more welcome clarity on the future relationship with the EU. "Important was that she stressed Britain's willingness to take over EU rules voluntarily in some sectors, in order to avoid disruption for business," he said.
Michel Barnier, the EU's chief negotiator, who held talks with May at 10 Downing Street Thursday, welcomed the speech, saying it provided "clarity" about the UK leaving the single market and customs union and a "recognition" that trade-offs will inform future talks on a deal.
The European Parliament's Brexit co-ordinator Guy Verhofstadt said May needed to move beyond vague aspirations, adding: "we can only hope that serious proposals have been put in the post."
Northern Ireland politician Arlene Foster, leader of the DUP, said: "The prime minister has set forward the basis upon which it would be possible to move forward. I welcome the prime minister's clear commitment that she will not countenance any new border being created in the Irish Sea between Northern Ireland and the rest of the United Kingdom. Securing a sensible outcome for everyone will require the EU27 to consider innovative solutions rather than rule out any proposal which has not been conceived in Brussels."
May said in her Mansion House speech she wants a UK-EU trade deal covering more sectors than any Free Trade Agreement anywhere in the world today.
"The world is watching," said May, "We should not think of our leaving the EU as marking an ending, as much as a new beginning for the United Kingdom and our relationship with our European allies."
"I am in no doubt that whatever agreement we reach with the EU, our future is bright," May concluded.