Prime Minister David Cameron has said he could have handled the row over his financial affairs "better", admitting it had "not been a great week".
Addressing the Tories' spring forum, he said he was to blame for the handling of revelations about his holding in his late father's offshore fund.
Days after questions were first raised, the PM admitted this week he had owned and later sold units in the fund.
Mr Cameron also said he would publish information on his tax returns later.
Labour leader Jeremy Corbyn said he "looked forward" to that publication.
BBC political correspondent Carole Walker said the decision to publish the tax returns would keep the prime minister's finances in the headlines.
But Downing Street felt the only way to move forward was by confronting the issue head on with an unprecedented level of detail, our correspondent said, adding however that Labour was in no mood to let the row drop.
Hundreds of protesters calling on Mr Cameron to "close tax loopholes or resign" rallied outside Downing Street on Saturday, some moving on to demonstrate outside the venue where the Conservative forum was held.
Addressing Conservative Party activists at the central London forum, Mr Cameron said: "It has not been a great week. I know that I should have handled this better, I could have handled this better.
"I know there are lessons to learn and I will learn them.
"Don't blame Number 10 Downing Street or nameless advisers, blame me."
'I want to be open'
Mr Cameron said: "I was obviously very angry about what people were saying about my dad. I loved my dad, I miss him every day.
"He was a wonderful father and I'm very proud of everything he did. But I mustn't let that cloud the picture. The facts are these: I bought shares in a unit trust, shares that are like any other sorts of shares and I paid taxes on them in exactly the same way.
"I sold those shares. In fact, I sold all the shares that I owned, on becoming prime minister."
Mr Cameron continued: "Later on I will be publishing the information that goes into my tax return, not just for this year but the years gone past because I want to be completely open and transparent about these things.
"I will be the first prime minister, the first leader of a major political party, to do that and I think it is the right thing to do."
Revelations about Mr Cameron's financial affairs followed a leak of 11 million documents this month held by Panamanian law firm Mossack Fonseca.
The documents, known collectively as the Panama Papers, revealed that Mr Cameron's late father Ian had been a client of Mossack Fonseca when establishing a fund for investors.
Labour's shadow chancellor John McDonnell has called for an "open and public" inquiry into the revelations.
BBC political correspondent Alan Soady
This mea culpa by the prime minister is solely on how the questions about his tax affairs have been handled, not on his decision to hold shares in his late father's offshore firm.
David Cameron remains adamant that he did not break any rules and that his father's Bahamas-based business had not been established for the purpose of avoiding tax.
But he is well aware that taking several days to confirm that he used to hold shares has allowed his political opponents to accuse him of "misleading" the public.
And so in an attempt to show he has nothing to hide, he will publish "information" from his tax returns.
That may or may not take some heat out of the row, depending on what the "information" actually is. But politically, a lot of damage has already been done.
When David Cameron refers to taking personal responsibility, he is seeking to shield his closest advisers in Downing Street, who some sections of the press - and even one or two Conservatives - have been criticising for the prime minister's flawed response.
Protesters at a demonstration in central London waved placards and chanted slogans criticising tax avoidance.
Downing Street protest organiser, journalist Abi Wilkinson, told BBC Radio 5 live the week's revelations raised questions about Mr Cameron's commitment to tackling tax avoidance.
"But the thing that really made us think we had to get out and protest was the news that, in 2013 when the EU were trying to crack down on offshoring and tax avoidance, he stepped in and actually weakened what they were trying to do," she said.
Meanwhile, Labour leader Mr Corbyn said the issue was, "not about an individual, not about one person, one family, it's about a whole ethos where the very rich are able to put their money into tax havens, offshore accounts".
"That is money that is being made that is untaxed and doesn't contribute anything to the need of public services."
Mr Corbyn said he would publish his own tax returns "very, very soon, when I've got the papers together".
Commons leader Chris Grayling said those accusing Mr Cameron of misleading the public were making a "mountain out of a mole hill".
What Cameron said when:
- Asked on Monday whether she could confirm that no family money was still invested in the fund, Mr Cameron's spokeswoman said: "That is a private matter"
- Then in an interview on Tuesday, Mr Cameron said: " I have no shares, no offshore trusts, no offshore funds, nothing like that. And, so that, I think, is a very clear description"
- Downing Street issued a statement later that day: "To be clear, the prime minister, his wife and their children do not benefit from any offshore funds. The prime minister owns no shares. As has been previously reported, Mrs Cameron owns a small number of shares connected to her father's land, which she declares on her tax return"
- No 10 then released a further clarifying statement on Wednesday, saying: "There are no offshore funds/trusts which the prime minister, Mrs Cameron or their children will benefit from in future"
- On Thursday the PM told ITV News: "We owned 5,000 units in Blairmore Investment Trust, which we sold in January 2010"
At the Conservative spring forum, Mr Cameron said local elections in England and mayoral contests in London, Bristol, Liverpool and Salford on 5 May gave voters a clear choice between "Tory competence and the disarray of the rest".
Only Tory councils could be trusted to keep taxes low while "getting things done" and "delivering more for less", he said.
The Conservatives are defending about 880 seats last contested in 2012.
More than 2,700 seats in 124 councils across England are up for grabs next month in what will be Labour leader Jeremy Corbyn's first national test in England at the ballot box.