The 77-page "white paper" was published a day after lawmakers voted overwhelmingly in favor of a bill that once finalized would give May the authority to invoke Article 50 of the Lisbon Treaty -- the formal process of leaving the EU.
• No "cliff-edge" Brexit. The government said it would seek transitional arrangements to minimize the impact of Britain's departure front he EU.
• Early targets for trade deals would be the US, China, Brazil, Australia, New Zealand, India and states in the Persian Gulf. It would seek "continuity" of arrangements with other countries that are currently covered by EU deals.
• The government hopes for an "early" agreement on the status of the 2.8 million EU nationals who live in the UK. It expressed disappointment that a number of EU states had blocked attempts to sort the issue out before formal divorce talks begin.
• It offered no guarantee on the status of the Irish border. The government said only that it would "aim" to protect the common travel area that exists between Northern Ireland, which is part of the UK, and the Republic of Ireland, which will remain in the EU. The issue is a key part of the deal that brought peace to Northern Ireland.
In a foreword to the white paper, May said Britain approached the negotiations anticipating success, not failure.
"The House of Commons has voted overwhelmingly for us to get on with it. And the overwhelming majority of people -- however they voted -- want us to get on with it too," she said.
David Davis, the Secretary of State for Exiting the EU, told the House of Commons that Britain wanted the "European Union to succeed, politically and economically" and so would approach the negotiations in the "spirit of goodwill" and for both sides' mutual benefit.
Immigration: 'Nobody will be thrown out'
Around 2.8 million EU nationals live in the UK, many of them Polish. Around 1 million UK nationals live in other member states, 300,000 of them in Spain.
Resolving the status of both groups was an "early priority", the white paper said. It indicated frustration with other EU states that it had not already been dealt with. "The government would have liked to resolve this issue ahead of formal negotiations," it said.
While welcoming the contribution of migrants to Britain's economy and society, the white paper said there had been "record levels of long-term net migration in the UK and that sheer volume has given rise to public concern." It cites pressure on housing, infrastructure, public services and downward pressure on wages for those on the lowest incomes.
"We will design our immigration system to ensure that we are able to control the numbers of people who come here from the EU," it says. "In future, therefore, the Free Movement Directive will no longer apply and the migration of EU nationals will be subject to UK law."
Pressed in the House of Commons by opposition Labour spokesman Keir Starmer, Davis said EU nationals would not be forced to leave the UK. "I am not going to be throwing people out of Britain and for him to suggest that is outrageous," he said.
Irish border: 'Aim' to keep current arrangements
Under the terms of the 1998 Good Friday Agreement, which brought an end to sectarian conflict in Northern Ireland, there is currently an open border between the north and south. Any change to this arrangement would imperil the peace process, critics have warned.
However, there are doubts that the UK could maintain an open border with an EU member state -- and the white paper acknowledges the doubts, saying only that the government "aims" to maintain the current arrangements.
Speaking last month, May said that finding a solution to the border issue would be "an important priority," adding that "nobody wants to return to the borders of the past, so we will make it a priority to deliver a practical solution as soon as we can."
Trade: 'No need to start from scratch'
The government will seek to retain a large part of the country's current trade arrangements with the EU, but emphasizes the need to boost trade with other states, citing growth in its exports to South Korea, China, Brazil and Mexico.
The paper reiterates May's recent announcement that Britain will not remain in the union's single market but will instead seek "the freest and most frictionless trade possible in goods and services between the UK and the EU."
But the document made it clear that the UK wanted to retain some parts of the present single market arrangements, saying it "makes no sense to start again from scratch" in areas where rules have existed for years.
The government will seek the freest possible trade in financial services, a key sector for Britain's economy.
Several prominent financial institutions based in London have suggested moving jobs to EU states to remain part of the single market. From within the EU, Britain has been able to provide financial services across the union under a common set of rules and regulatory authority -- access known in the sector as "passporting."
More than 5,000 UK firms use passporting to provide services to the EU, while around 8,000 European firms use the instrument to provide services to Britain, and UK-based financial institutions have lobbied the government to try to retain passporting rights.
The government hopes to seek new opportunities and reform policy in the agriculture, food and fisheries sectors, which the paper described as "currently heavily influenced by EU laws."
Security: 'Deepen security cooperation'
The government says it will continue to work with the EU in fighting terrorism and maintaining security, adding it is "uniquely placed to develop and sustain a mutually beneficial model of cooperation."
It says that the UK is "driving or co-driving" approximately half of the Europol projects against organized crime, and cites the extradition of more than 8,000 individuals accused or convicted of a criminal offense to other EU countries under the European Arrest Warrant.
It reiterates its willingness to work with EU allies over criminal matters, security and cyber security while upholding its commitments to NATO.
It also highlights that it will "seek a strong and close future relationship with the EU with focus on operational and practical cross-border cooperation."
Transition: 'No cliff-edge'
The government says it wants to deliver a smooth, orderly exit and to "avoid a disruptive cliff-edge" as Britain leaves the European Union.
The government envisages finalizing a deal within the expected two-year period after triggering Article 50, but seeks a phased transition after that, to maintain stability and give businesses and institutions time to adjust.
It does not give a timeline for how long the whole process might take, saying some elements will be phased in faster than others. But it stresses Britain will not seek an "unlimited transitional status" which would be good for neither side.