Of 27 bills, eight relate to Brexit and its impact on immigration, trade and sectors such as fisheries and farming.
Prime Minister Theresa May urged MPs to "seize this moment of national change" to unite and work for a fairer country.
But Labour leader Jeremy Corbyn said she had lost authority after axing a raft of manifesto pledges.
Proposals to scrap the winter fuel allowance for well-off pensioners, end automatic 2.5% pension rises, expand grammar schools and end free school lunches for all infants have been dropped, while reforms to social care funding will be put out to consultation and a cap on some energy tariffs considered further.
Amid continuing talks with the Democratic Unionists about them supporting Theresa May's government, a Downing Street spokesman said it was confident the Queen's Speech could "command the confidence" of the House of Commons in a vote next week.
The main non-Brexit proposals include:
There was no mention of US President Donald Trump's proposed state visit to the UK later this year, appearing to confirm suggestions it has been delayed. Ministers said the reason it was not included was purely because no date had been set.
The Queen announced the government's legislative programme for the next two years at the State Opening of Parliament.
She was accompanied by the Prince of Wales, rather than the Duke of Edinburgh, after Prince Philip was admitted to hospital on Tuesday night. Buckingham Palace said it was a "precautionary measure" for treatment of an infection arising from a pre-existing condition.
By political editor Laura Kuenssberg
It was meant to be Theresa May's political coronation, but the Queen's Speech has confirmed the reality of her fall from grace.
The prime minister's ambitions for significant change at home have been culled, disappearing with her majority.
But the complexity of all the work the government has ahead administratively, as the UK prepares to leave the EU, is plain to see.
Eight bills on Brexit each requires no less than a redesign of systems that have been in place for decades. Each will require careful political handling, at a time when the government cannot be sure of its majority and a Labour Party with wind in its sails is determined to be a guerrilla opposition. Read more
As MPs began debating the government's plans, Mrs May said the country was split "between red and blue, young and old and Leave and Remain" and Parliament's challenge was to heal, not reflect those divisions.
She promised to work with "anyone in any party" in the national interest on Brexit and other issues.
"Not every problem can be solved by an act of Parliament but it is a step forward to building a more compassionate, united and confident nation," she said.
She also apologised for "the failure of the state, national and local" in its response to the Grenfell Tower fire, promising a new role of independent public advocate to represent bereaved families in the aftermath of disasters.
With Brexit talks now under way, the government has set out the laws needed to leave the EU - irrespective of the final deal agreed with Brussels.
At the heart of this is the so-called Repeal Bill - which will repeal the 1972 European Communities Act and end the jurisdiction of the European Court of Justice. It will also copy over all EU laws into UK law, with Parliament then deciding which bits to retain.
The government says "wherever practical the same rules and laws will apply after exit, therefore maximising certainty for individuals and businesses".
The bill would give the Parliament temporary authority, via secondary legislation, to amend laws that do not "operate appropriately" after Brexit while existing decision-making powers devolved to Scotland, Wales and Northern Ireland will be maintained pending further discussion on a permanent solution.
As an indication of the scale of change which Brexit will bring, seven other pieces of legislation are proposed to anticipate the end of EU jurisdiction and introduce national policies in key sectors.
On immigration, a bill will legislate for the end of free movement from the EU and make the status of EU nationals and family members subject to UK law. Although there are no specific details about a new system, ministers say they will be able to "control" numbers while attracting the "brightest and the best".
A Fisheries bill will allow the UK to take on responsibility for "access to fisheries and management of its waters" while an Agriculture Bill will "provide stability" for farmers and ensure an "effective system" of support to replace the Common Agricultural Policy.
A new nuclear safeguards regime will be required after the UK leaves the EU and its nuclear agency Euratom, with new powers for the Office for Nuclear regulation.
Other measures will allow for a standalone domestic customs regime, giving the UK the scope to make changes to VAT and excise rates currently determined by the EU, to pave the way for an "independent trade policy" and to enable the UK to implement non-UN sanctions on its own or in conjunction with allies.
The government has cancelled next year's Queen's Speech, so this one will cover a two-year period to give MPs more time to debate all the Brexit legislation.
The remaining 19 bills - including three in draft form and three finance bills - are a mixture of new proposals and legislation carried over from the last Parliament, which was cut short by the snap election.
Among proposals that will not require immediate legislation, the government is to review its counter-terrorism strategy in the wake of recent attacks in London and Manchester and establish a new Commission for Countering Extremism to "stamp out extremist ideology in all its forms".
While there are no proposed full laws on health and education, a review of mental health legislation is planned while a "digital charter" will seek to boost online safety and digital commerce.
Labour is putting forward an alternative version of the Queen's Speech, calling for an end to austerity and huge investment in public services.
Mr Corbyn called it a "threadbare legislative programme from a government that has lost its majority and apparently run out of ideas altogether".
He said Labour would "use every opportunity to vote down proposals which do not have public support".
The Lib Dems said the Queen's Speech was "bereft" of ideas to support the public services while the Green Party said it was a "shell of a Queen's Speech from a hollowed out government".
The CBI said there had been a "welcome change of tone" towards business but ministers should put "pragmatism before politics" over Brexit. The TUC said promises to help working people were "vague".
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A host of proposed new laws designed to prepare the UK for a "smooth and orderly" departure from the EU have been announced in the Queen's Speech.