WASHINGTON—President Biden sought to rally public support for a multitrillion-dollar education and child-care plan at the core of his domestic-policy agenda, as Democrats in Congress wrestle with disagreements over the size and scope of the package.

“To truly win the 21st century and once again lead the world, to truly build an economy from the bottom up and the middle out, to truly deal everybody in this time, we need to invest in our people,” Mr. Biden said at McHenry County College in Crystal Lake, Ill.

The president is simultaneously pressing Congress to pass a $1 trillion bipartisan infrastructure bill and the education and child-care plan, which Democrats hope to approve without Republican support through a process tied to the budget known as reconciliation.

Both measures face challenges. The White House needs at least 10 Republican votes in the Senate to pass the infrastructure package, which was negotiated by a group of centrist senators. And Democrats must stick together if they want to move the education and child-care plan without the backing of Republicans. Democrats can lose no more than four Democratic votes in the House and none in the Senate on legislation opposed by all Republicans.

Mr. Biden pitched his plan in a county he lost in the 2020 election to former President Donald Trump, though Mr. Biden carried the state. The suburb north of Chicago is part of a congressional district represented by Democratic Rep. Lauren Underwood, who ousted a GOP incumbent in 2018 and is now being targeted by Republicans hoping to reclaim the House majority next year.

The president’s speech came after two federal agents and a Chicago police officer were shot during an undercover operation Wednesday, following a deadly holiday weekend where 100 people were shot and 18 were killed. During a meeting at the airport with Chicago Mayor Lori Lightfoot, Mr. Biden said he would work with city officials to stem gun violence.

Mr. Biden has proposed a raft of antipoverty and family-focused measures that he hopes will be included in the Democrats-only legislative package. Among them: tuition-free community college and prekindergarten; federal funding for affordable child care and housing; paid family and medical leave; expanded home care for elderly and disabled Americans; and measures to boost renewable energy.

Republicans are largely opposed to his proposals, warning against raising taxes and trillions in additional spending, and arguing that the federal government shouldn’t be playing such a central role in Americans’ lives. GOP lawmakers have also contended that the White House has expanded the definition of infrastructure too much to include what Mr. Biden often calls “human infrastructure,” such as education, workforce development and efforts to combat climate change.

In his roughly 30-minute speech, Mr. Biden said his policy proposals would boost the economy and help low- and middle-class Americans.

“The last time the economy grew at this rate was in 1984, when Ronald Reagan was telling us it was an American morning,” Mr. Biden said. “Well, this is going to be an American century.”

Mr. Biden’s speech was packed with details about tax credits and funding levels. “I know that’s a boring speech, but it’s an important speech,” he said.

Democrats haven’t yet agreed on how much the package should cost; nor are they aligned on how much of it should be paid for. Mr. Biden has proposed funding his proposals with taxes on the wealthiest and highest-income Americans.

Sen. Joe Manchin (D., W.Va.), who has said that the legislation should be fully paid for, has said he could support a package of around $2 trillion. Senate Budget Committee Chairman Bernie Sanders (I., Vt.) has floated spending as much as $6 trillion, with half of it paid for by tax increases.

Many congressional Democrats are using the fact that their party’s leaders can lose so few votes as leverage to issue demands for the budget package. Lawmakers from New Jersey, New York and California have pressed to roll back the $10,000 limit on the state and local tax deduction, while others in the Midwest are pushing for investments in biofuels.

Rep. Chuy García (D., Ill.) said in a statement Tuesday afternoon that any budget package must include a pathway to citizenship for Dreamers—young unauthorized immigrants brought to the country as children—and farmworkers, among other groups of immigrants.

“I can’t support any deal that leaves so many people in my district behind,” he said in a statement.

Mr. Sanders has said he hopes to include some immigration provisions. However, lawmakers have said it isn’t clear if they will pass muster with the Senate parliamentarian, who issues guidance on what is permitted under the special rules Democrats hope to use to pass the measure without Republican votes. To comply, a measure must be determined to have a meaningful fiscal impact that can’t be “merely incidental” to the policy proposal.

Write to Andrew Restuccia at andrew.restuccia@wsj.com and Kristina Peterson at kristina.peterson@wsj.com