MANCHESTER, N.H. — A vast array of Democratic leaders, divided by generations but uniformly emboldened by President Trump’s perceived vulnerability, have begun taking palpable steps toward seeking the White House in an election that is still three and a half years away.

A prominent member of that group, former Vice President Joseph R. Biden Jr., may have sounded a starting gun for the 2020 race on Sunday night as he delivered a tough-love manifesto for the party’s future to a crowd of Democrats here.

Mr. Biden called on Democrats to deliver an optimistic message aimed at the needs of struggling Americans and to emphasize issues like health care and education, which he described as largely sidelined in the last election. “I’m absolutely positive they want to be with us,” he said of people in economic distress. “We have to prove, again, we understand that hopelessness.”

Mr. Biden said he was prepared to storm the country to help revive the party, and boasted of his comfort in wooing areas that voted for Mr. Trump. “Wherever you want, just let me know, because it’s important,” he said.