The First Minister said Scotland's written constitution should also outlaw possession of nuclear weapons and lay down rules for the use of the armed forces in global conflicts.
Mr Salmond set out his vision for a formal constitution in a speech in London.
Under the SNP's plans, Holyrood would start to draw up the document after the 2016 election if Scots vote for independence in the 2014 referendum.
Mr Salmond said all political parties and citizens would be encouraged to contribute their views through a constitutional convention.
He said the ancient Scots principle of sovereignty of the people – people's right to decide their own form of government – should be enshrined in the constitution.
The SNP leader added: "In Scotland, we have a policy of the right to free education in keeping with our history as the nation which pioneered universal education.
"We also have homelessness legislation which is proving effective by granting rights to people who are made involuntarily homeless. There is an argument for embedding those provisions as constitutional rights.
"A second key issue which the constitution could examine could include the future of Trident. A constitutional ban on the possession of nuclear weapons would end that obscenity.
"And, thirdly, is the issue of the use of our armed forces. In 2003, the Westminster Parliament was effectively misled into sanctioning the illegal invasion of Iraq. We should therefore explore what parliamentary and constitutional safeguards should be established for the use of Scottish forces in international engagement."
Mr Salmond said the UK's lack of a written constitution was a "democratic deficit that an independent Scotland should not repeat".
He insisted a written constitution was one of the key arguments for independence and drawing up the list of basic rights would be "one of the most fundamental and exciting" tasks of the first post-independence parliament.
Labour MSP Paul Martin said: "Rather than a new constitution perhaps the Government could do what it claims to.
"Rather than a constitutional right to a home, perhaps the Scottish Government could properly fund housing and tackle homelessness the way Scots law demands.
"Rather than a constitutional right to education, the Scottish Government could commit to funding education properly so college places aren't slashed and people have proper access to education."
However, he added: "Perhaps we could have a duty for the First Minister to tell the truth – so the First Minister doesn't claim he has legal advice when he hasn't even sought it."
Scottish Conservative leader Ruth Davidson MSP said: "This is little more than political posturing from Alex Salmond. Instead of grandstanding in London, he should be getting on with running the country and using the powers he already has to help improve the daily lives of the people of Scotland."
However, the plan was welcomed by Canon Kenyon Wright, who chaired the constitutional convention which paved the way for the Scottish Parliament. He said: "A constitution defines and safeguards the values and character of the nation, as a participative democracy based on social justice, equal law, and the sharing of power."
Scottish Greens leader Patrick Harvie said: "The Scottish Greens strongly support a constitutional bar on nuclear weapons and on nuclear alliances, and given the chance to shape a Scottish constitution we would make that case."