COLUMBUS, Ohio — Pressing a contrast with his rivals, President Obama expanded his critique of the Republican budget blueprint Tuesday to focus on education policy, arguing that Mitt Romney would reverse his administration’s policies to aid education despite the consequences to the economy.

For the president, who addressed a crowd of about 3,300 people at Capital University on the eastern edge of Ohio’s capital city, the policy wasn’t just about politics. It was also personal.

Drawing on his and his wife’s struggles to pay off student debts, Obama said at the outdoor rally that he was familiar with the difficulties of paying for college.

“We’ve been in your shoes,” he said. “I’m only standing before you because of the chance that my education gave me. So I can tell you with some experience that making higher education more affordable for our young people — it’s something I’ve got a personal stake in.”

Democratic officials call education “one of the most important economic issues facing our nation.” The stop was the first of three in the next days at either college or high school settings.

The president’s campaign is eager to highlight how a budget proposal offered in Congress by Rep. Paul D. Ryan, the GOP vice presidential pick, would affect key areas beyond Medicare, the focus of initial skirmishing after Romney added the Wisconsin lawmaker to the ticket. But the campaign is also seeking to reignite enthusiasm among younger voters, like the ones present at the rally and the ones he interacted with at an earlier unscheduled stop on the campus of Ohio State University.

The Romney campaign responded to Obama’s focus on education by saying that under his leadership, “too many young Americans are suffering from higher college costs, more debt, and a lack of good jobs when they graduate.”

“Today’s policies are just more of the same from a president who hasn’t fixed the economy or kept his promises to the young people who supported him four years ago. The Romney-Ryan plan will deliver 12 million new jobs to help recent graduates — and all Americans — enjoy a more prosperous future,” spokeswoman Amanda Henneberg said.

The Obama campaign argued that Ryan’s budget, which Romney has said is similar to his own proposal, would require spending cuts that would gut many of the proposals Obama has pushed through.

The president in his remarks never mentioned Ryan by name, though he referred to congressional Republicans more broadly and alluded to what he said was Ryan’s plan to “voucherize Medicare.”

But his main focus was Romney. He spoofed comments the Republican made at Ohio events — including one in which he said students who couldn’t afford college should “shop around,” or that they could borrow money from their parents.

“Not everybody has parents who have the money to lend. That may be news to some folks, but it’s the truth. So what Gov. Romney’s offering us is not an answer,” he said.

Obama said he hadn’t just “talked the talk,” but “walked the walk” on education, boosting Pell grants, tying student loan payments to incomes, among other policies.

He urged students to make sure they registered to vote, at the very least. Republicans “know their economic plan isn’t popular.... So they’re betting on the fact that you get so discouraged that you decide that your vote doesn’t matter.”

“Young people especially, I’m going to need your help,” he said. “Let’s show them your voice makes a difference.”

The education push has added resonance in Nevada, where Obama traveled later Tuesday for a speech at Truckee Meadows Community College in Reno. Earlier this month the Obama administration approved Nevada’s request for a waiver from No Child Left Behind, making it the 33rd state to receive one.

On Wednesday, he will round out the two-day campaign trip with an event at a North Las Vegas high school, before traveling to New York for fundraising events.

michael.memoli@latimes.com