After winning the biggest landslide since Chile's return to democracy, the president-elect, Michelle Bachelet, vowed on Monday to push ahead with an ambitious programme of tax, educational and constitutional reforms to address inequality.

The centre-left politician – who secured 62% of Sunday's vote – was also expected to propose legislation on reproductive rights and same-sex marriage in this predominantly Catholic nation.

In a switch of power, Bachelet trounced her conservative opponent, Evelyn Matthei from the Alianza coalition, which has run the country for the past four years.

Despite a low turnout of 42%, the win puts the Chilean Socialist party leader back in the La Moneda presidential palace, where she had been the incumbent from 2006 to 2010. Her first administration was popular, but made only modest inroads into reducing inequality. In her second term, the president-elect has promised more radical changes.

"Chile has looked at itself, has looked at its path, its recent history, its wounds, its feats, its unfinished business and this Chile has decided it is the time to start deep transformations," Bachelet told supporters in a jubilant victory speech.

Thousands gathered to celebrate at the San Francisco church on La Alameda, a wide avenue in central Santiago, which echoed with cumbia music on Sunday night.

Alongside the red, white and blue of the Chilean flag there was a riot of other colours: the red of the communist party and the multicoloured flag of the indigenous Mapuche movement. Bachelet's Nueva Mayoría bloc embraces a broad coalition of centre-left parties.

Education is high on the 62-year-old president-elect's agenda. Bachelet has promised that the state will pay the tuition fees of the poorest 70% of Chile's higher education students. She is also expected to create more public universities to break the monopoly of private institutions.

"[These reforms] are about making sure that the education system doesn't reproduce an unequal society," said Ernesto Ottone, a professor at Santiago's Diego Portales University.

Four student leaders were elected to Congress this year, something which seemed impossible when students began demonstrating for better quality and more affordable education almost three years ago. Whether they support the new president will depend on her actions.

One of the student MPs, Gabriel Boric, told the Guardian: "If Michelle Bachelet responds to the expectations she has created on education we are going to support her but we are not so sure that the people around her believe in what we believe, so there will be tension."

To fund its education policies, the new administration aims to steadily raise corporate taxes from 20% to 25%, which would bring Chile closer to the norm among developed nations. This may risk a backlash from international investors, which benefited from the more business-friendly policies of the outgoing president, Sebastián Piñera, but the country is moving in a new direction.

"It's the end of a political and economic cycle and the beginning of a new one. We have a new agenda in our country," said Ricardo Lagos, Chile's president from 2001-06. Income inequality was the first issue Bachelet must face, he said.

Income per head in Chile has grown from around about $4,500 (£2,800) a year in 1990 to more than $20,000 in 2013. But of the 34 countries of the Organisation for Economic Co-operation and Development (OECD), Chile has the biggest gap between rich and poor.

The extent to which Bachelet can meet expectations for change will depend on the economy, which is slowing down as a result of US financial policy and lower demand from China, and political negotiations in Congress.

Bachelet's bloc won a majority in legislative elections last month, but it will need to win over independent and opposition legislators if it is to pass educational reforms that require a 57% share of the vote and constitutional changes that need a 67% majority.

Before the election, Bachelet said she wanted to change the dictatorship-era constitution. "They didn't believe in democracy so they built a system that meant even if people voted for change, it was hard to alter policy," she said. "We need a constitution born in democracy. The one we have now is illegitimate."

She is proposing revisions to Chile's unique binomial electoral system, which was introduced by the former dictatorship of Augusto Pinochet to stifle democratic change, the high-majority requirement for educational reforms, and the constitutional court, which – very unusually – has preventative powers to withdraw bills that judges consider unconstitutional.

Conservative politicians have already stated their opposition. Felipe Morandé, former campaign manager for Evelyn Matthei, the right-wing candidate who stood against Bachelet, said: "[Changing the constitution] will cause more uncertainty, which means there's going to be a less favourable climate for business, and then less investment and less growth and less increases in wages and jobs."

Other planned constitutional changes would see the inclusion of rights for women and indigenous groups.

Currently, there is strong opposition in the Catholic church – an influential force in Chilean politics – to wider abortion legislation. But Bachelet – a paediatrician pediatrician by training and a former head of United Nations Women – is a firm believer in women's sexual reproductive rights and says she would like a broader public discussion on this topic.

"I think women should make their own decisions on sexuality and reproduction," Bachelet said. "But a president can't impose views on society ... I think we have to discuss this."

A more controversial proposal is to extend term limits. Currently, a president may only serve one four-year term. After this, they have to sit out a term before they can stand again. Bachelet said she wanted to either extend the length of a single term or allow for one re-election. This may raise fears that she is following hard-left leaders in the region, such as Hugo Chávez in Venezuela, Rafael Correa in Ecuador and most recently Daniel Ortega in Nicaragua in strengthening her personal power. Bachelet adamantly rejects this claim and says that she – like Brazil's former president Luiz Inácio Lula da Silva – bucked this trend in Latin America.

"There is no chance of that. It tempts many, but I'm not tempted," she said.

Chile's economy grew by 5.6% last year but the growth rate is expected to ease to 4.2% this year and the central bank warns it could drop below 4% in 2014. Patricio Navia, a political scientist at New York University, said: "Chile has grown very rapidly over the last four years. That's not going to happen next year. Unemployment will increase and as soon as that sets in people will be more concerned about employment than about free education and the other promises that Bachelet has made."

Voters said they expected significant but gradual change.

Gloria Salas, at a polling station in Santiago, said: "I think that she can make the changes I want without being really radical: A new constitution, health, education, pensions.

Ellerman, a former student who voted on Sunday and preferred not to give his last name, said: "I left university a year ago and I'm paying off my loan. I think it's really important that people can study for free. It's going to be really difficult to make it 100% but little by little it can be done."