Biden has a second-term wish list. Congress could spoil his plans.

But if Democrats don’t expand their numbers on Capitol Hill, many of those plans could be dead on arrival in Congress, suffering the same fate they faced in Biden’s first term. That will raise the stakes as Democrats fight to retake the House and hold their majority in the Senate in order to enact many of Biden’s biggest plans.

Lowering costs for Americans would be a theme of a second term, according to Biden’s aides, who said they believe that will resonate with 2024 voters who consistently cite inflation as one of their top concerns.

The president wants to pass legislation expanding to all Americans the $35 cap on out-of-pocket costs for insulin. The cap, which was included in a healthcare, climate and energy law that Congress passed in 2022, currently applies to Medicare recipients. The president’s advisers are discussing a range of additional healthcare-related proposals aimed at expanding on the administration’s efforts to lower prescription drug prices and limit other medical expenses.

Also expected in a second term: a push to revive elements of Biden’s Build Back Better agenda that were rejected by Republicans and some moderate Democrats, including free prekindergarten; subsidized child care; paid family and medical leave; and expanded care for elderly and disabled Americans.

Biden would continue his longstanding campaign to change the country’s gun laws, including an assault-weapons ban and universal background checks. He would push Congress to codify Roe v. Wade into law. And he would lobby for the passage of legislation to preserve and expand voter access as many GOP-led states impose laws tightening voting rules. Those measures would face opposition from Republicans.

Biden will continue to push Congress to pass legislation to crack down on so-called junk fees, including charges for online ticket purchases, seating on commercial airplanes and vacations at resorts. His team is also weighing new measures to reduce student loan debt, tackle climate change and address the high cost of housing, aides said.

To pay for his wide-ranging plans, Biden would continue his push to raise taxes on the wealthiest Americans. He has proposed a “billionaire minimum income tax," which would require that households worth more than $100 million pay at least 20% in tax on their income and rising asset values each year. The president has called for raising the top individual income-tax rate to 39.6% from 37%, quadrupling the 1% tax on stock buybacks and increasing the corporate tax rate to 28% from 21%. The president is also pushing for restoring the expanded child tax credit, which expired in 2022 and sharply cut child poverty.

How Biden campaigns on his second-term agenda has become a point of tension as Democrats prepare for the November elections. Some prominent progressives want him to lay out his plans in more detail and draw a sharper contrast with his likely Republican challenger, former President Donald Trump, who has outlined dozens of policies to advance populist conservative priorities. Biden campaign officials are instead holding back some details of the incumbent’s second-term plans, arguing that voters will be more receptive to them when the election gets closer.

“I think we need to be clearer," said Rep. Ro Khanna (D., Calif.), calling on Democrats to sharpen how they talk about Biden’s agenda, “and we also need to give [voters] a sense of…how achieving it requires building majorities in the House and the Senate."

Amid polling that shows widespread dissatisfaction with the economy, Biden should acknowledge that voters are struggling with the cost of college, child care, healthcare, housing and other necessities, Khanna said. “Then the president should say, ‘I have very concrete plans’" to address those issues, he said. Khanna’s message echoes that of other Democrats, who have raised similar concerns.

“I know that some prices are still too high for too many," Biden wrote on X on Sunday. “I am doing everything in my power to lower costs—from energy bills and medicine to addressing hidden junk fees companies use to rip you off."

Still, many of Biden’s speeches have focused on selling what he accomplished in his first three years in office. The president hasn’t yet devoted a speech specifically to laying out a second-term vision.

The president’s advisers argue there is plenty of time to detail his agenda and they contend that Biden’s priorities are no secret. They note that their approach mirrors past incumbent presidents who waited until later in the year to explain their second-term visions. Biden’s aides believe that most voters won’t begin seriously engaging with the coming election until this spring or summer, and they are planning key campaign rollouts around that timeline.

“Still 10 months away from an election most Americans are not yet focused on, we look forward to scaling our efforts across the board this year to mobilize our coalition, including speaking to them more about how a second Biden-Harris term will deliver on important issues for them at the right moment to maximize impact," Biden campaign spokesman Kevin Munoz said.

As the campaign heats up, Biden is planning to increasingly make the case that Trump’s policy proposals would favor the wealthy and large corporations, and that re-electing him would threaten the foundations of the country’s democracy. “Donald Trump’s campaign," Biden said on Friday, “is about him, not you." A Trump campaign spokesman didn’t respond to a request for comment. The former president, in a speech in Iowa last week, accused Biden of fear mongering.

Biden’s State of the Union speech, which is scheduled for March 7, will be the president’s highest-profile opportunity to lay out a forward-looking vision for this year and beyond. The televised address typically captures the attention of millions of Americans.

The president has called on lawmakers to focus on issues that could win bipartisan support on Capitol Hill, and he is expected to echo that message in his coming address. They include strengthening privacy protections for children, boosting mental health services, curbing fentanyl, helping veterans and making advancements in the treatment of cancer.

Biden and his senior advisers don’t want to appear as though they are giving up hope of making progress on key policy issues this year. White House officials said they don’t differentiate between first- and second-term policy priorities and are eager to enact the president’s plans as soon as possible.

On some issues, the White House already has reached the limit of what it can achieve without Congress. But White House deputy chief of staff Bruce Reed said the president would continue to flex his executive authority whenever possible. Reed also acknowledged that Biden needs the House and the Senate on his side.

“Last year was one of Congress’s least productive years in memory," Reed said in an interview. “So we’ll need a more productive Congress, to say the least."

Write to Andrew Restuccia at andrew.restuccia@wsj.com