Writing for the Sunday Times after the re-election of Labour leader Jeremy Corbyn, Mr Jones said trolling and threats within Labour had to "stop".
But he said Labour rebels would be awarded "no prizes" for sulking.
Meanwhile he revealed UKIP and the Tories would attend a Welsh Government Brexit advisory group.
They will join representatives of all parties elected to the assembly.
Mr Jones is set to speak at the Labour party conference on Sunday.
The first minister wrote that Jeremy Corbyn's second win "gives us all some certainty in uncertain times. Every Labour Party member now has a responsibility to show our best selves to the rest of the country".
He said: "The battle for the soul of post-Brexit Britain won't be won by a paranoid political movement more concerned with rooting out 'traitors' than it is with the economy, security and happiness of our country.
"The booing, the hissing, the name-calling, the trolling, the threats… it has to stop.
"In what world are we living in, when the Labour party has to commission a report into anti-Semitism in our own party?"
Mr Jones said there needed to be "recognition now that no battle of ideas can ever be truly won, that challenge and difference can be channelled into a successful and united party".
"It needs some give and take. Just as those of us who campaigned for Remain will be awarded no prizes for sulking post-Brexit, those who challenged Jeremy will be awarded no prizes for sulking after this weekend," he said.
"It is time to get on with it."
BBC NEWS Reports
BBC NEWS Reports