WASHINGTON  In Congress and in statehouses, Republican lawmakers and governors are claiming a broad mandate from last year’s elections as they embark on an aggressive campaign of cutting government spending and taking on public unions. Their agenda echoes in its ambition what President Obama and Democrats tried after winning office in their own electoral wave in 2008.

In Washington, the House approved more than $60 billion in spending cuts before dawn on Saturday that would hit virtually every area of government, setting up a showdown with Senate Democrats and the White House.

In New Jersey, Gov. Chris Christie, who is embroiled in a political fight with the teachers’ union, is expected to call for drastic steps on Tuesday to close a roughly $10 billion budget gap.

And nowhere has a newly elected Republican flexed more muscle or drawn a more severe backlash than in Wisconsin, where Gov. Scott Walker is pushing to curtail collective bargaining rights for state employees. “For us, this is about balancing the budget,” Mr. Walker said on “Fox News Sunday.” “We’ve got a $3.6 billion budget deficit. We are broke. Just like nearly every other state across the country, we’re broke.”