The speech indicated how Mr. Biden will seek to bring his “working-class Joe” brand to a more diverse Democratic electorate, which is attempting to balance a pragmatic desire to beat President Trump with a strong tide of progressive energy animating the primary campaign. In the crowded Democratic field in which candidates are searching for ways to distinguish themselves, Mr. Biden is focusing on Mr. Trump, casting him as an outlier president in need of an emergency course correction.

Black voters play an outsize role in the Democratic electorate in South Carolina, and Mr. Biden’s high favorability ratings among that demographic are partly why he is considered the primary’s early front-runner. The Saturday rally, with a roughly 60 percent white crowd, opened with a performance from a local youth gospel choir and marching band. In his speech, Mr. Biden repeatedly emphasized his eight-year tenure as vice president to Mr. Obama, the country’s first black president.

“He’s a hell of a guy,” Mr. Biden said at one point.

“My buddy,” Mr. Biden said at another, before stopping himself. “My buddy? I shouldn’t be so casual. The president of the United States, Barack Obama.”

Mr. Biden has forged deep relationships in the South throughout his decades-long political career and arrived at the Hyatt Park gymnasium to whoops and cheers, a testament to how much good will remains even after he delayed his presidential announcement for months. Mr. Biden and his wife, Jill, are set to attend a private fund-raiser in Columbia on Saturday evening, and a church service in West Columbia on Sunday morning.

Missing from his speech were two high-profile elements in Mr. Biden’s career that are sure to follow him throughout the campaign and that could complicate his cozy relationship with black voters. This year Mr. Biden expressed regret for the crime and criminal justice legislation he championed in the early 1990s, saying it resulted in unequal sentencing and burdened black communities. Mr. Biden has also faced scrutiny for his handling of the 1991 confirmation hearings of Justice Clarence Thomas, who was accused of sexual harassment by Anita Hill.