Mr Smith, who joins Angela Eagle in challenging Jeremy Corbyn, said the tax system needed to be more "progressive".
On Brexit, he said people should not accept "we're on a definite path out."
Mr Smith and other Labour MPs say there should be a "unity" candidate to take on Mr Corbyn, who has vowed to fight the challengers in the contest.
But Ms Eagle, who was the first to launch a challenge against the Labour leader, said it was "too early" to talk about that idea.
Meanwhile, Mr Corbyn told the BBC's Sunday Politics he thinks the rules which exclude recently signed-up Labour members from voting in the contest are "not very fair" - and that he wants the party's National Executive Committee to change them.
He also believes the £25 fee for registered supporters to vote in the contest is too high.
In a speech in his constituency later, Mr Smith, MP for Pontypridd, will set out his leadership pledges, including plans to invest £200bn into building projects, which he will say is what is needed to "rebuild Britain".
Speaking on Sunday's Andrew Marr Show, Mr Smith, when asked if he would raise taxes on the richest in society, said: "I think we need to completely overhaul our tax system, so yes."
The former shadow work and pensions secretary said he would reintroduce a 50p top rate of tax "tomorrow" and also said it was "completely anomalous" for capital gains tax to be 20% when the higher rate of income tax was 45%.
Mr Smith, a former member of the Campaign for Nuclear Disarmament (CND), went on to say he would vote to renew Trident in Monday's Commons vote, saying he believed the world had become "more volatile, more insecure over the last few years".
"I want a world without nuclear weapons altogether, but I don't think we hasten that by divesting," he said.
Asked whether he would be prepared to press the nuclear "button" as prime minister, Mr Smith said: "You've got to be prepared to say 'yes', that's absolutely right."
BBC political correspondent Alan Soady
Both Angela Eagle and Owen Smith are putting themselves forward as anti-Jeremy Corbyn candidates and as candidates of unity who think they're not as far left as Mr Corbyn - but are not Blairites either.
By positioning themselves in that way, they think they could potentially be acceptable to both wings of the party.
The problem is that having two rival unity candidates splits the vote and one would have thought that it makes it more likely that Jeremy Corbyn will win this leadership contest.
Owen Smith has already said he believes there should only be one of them going forward into the formal contest - it's just a case of deciding who.
On the EU, Mr Smith hinted at the possibility of a second referendum on EU membership, but said it would depend on what kind of deal the UK negotiated on the terms of Brexit.
He also said it would be "very tempting" for Labour to campaign on a promise not to trigger the formal Article 50 process to bring the UK out of the EU, "because we are a party that believes in being at the heart of Europe".
"I don't think we should accept we're on a definite path out. I think we need to make sure people are satisfied," he added.
"We trusted people, rightly, to take the decision. We can trust them again in 18 months time to check it's absolutely what they wanted."
Mr Smith will further warn that the party risks "falling apart" if it does not act "like a proper team", and he is also expected to attack Mr Corbyn for being anti-austerity without putting forward an alternative.
In her pitch for the Labour leadership, Ms Eagle said she was a "working-class woman" from the North of England, "and that's what we need at the moment".
"My whole political mission is to get working class kids the right opportunities to shine," she said, and highlighted her experience in government and opposition.
The ex-shadow business secretary also urged Labour members and supporters to "keep it comradely" in the leadership election, following abuse and threats directed at MPs.
Labour Party members, affiliated trade union supporters and so-called registered supporters are able to vote although there are some key differences from the 2015 contest, which Jeremy Corbyn won:
Mr Smith has said he believes only one challenger to Mr Corbyn should appear on the final leadership ballot, decided by "whoever is the person who commands the largest degree of support in the PLP".
But Ms Eagle - who was debating the leadership contest with Mr Smith on the Marr show - said she thought she was most likely to beat Mr Corbyn, adding: "We're not going to do a deal here on your sofa."
It comes ahead of a parliamentary Labour Party hustings on Monday, when the leadership contenders will set out their stall to fellow MPs.
Yvette Cooper - who stood for the Labour leadership last time round - endorsed the idea of a single candidate to avoid "artificial rows".
She said an early general election was a possibility, and added: "So we need a unity candidate who can pull a strong team together, something that Jeremy Corbyn's not been able to do."
Despite facing a revolt from his MPs, Mr Corbyn retains the strong support of many party members and has said he will fight the challenges in a contest which is expected to be decided in September.
Labour has been "too timid" about taxation, leadership hopeful Owen Smith has said, and pledged that he would raise the top income tax rate to 50p.