A week after he effectively clinched the Republican nomination, Mitt Romney is turning to battleground states to pitch himself as the person to reinvigorate the U.S. economy while highlighting what he calls President Barack Obama’s “record of failure.”

Stepping beyond states holding GOP primaries, Mr. Romney is spending the week in a march through states that will be crucial to his bid to beat Mr. Obama in the general election—Pennsylvania, North Carolina, Ohio and wrapping up in Arizona.

But even as polls find voters more approving and ready to give him a second look, the former Massachusetts governor has unfinished business to do both in laying out a governing agenda and shoring up support among conservatives who strongly oppose Mr. Obama but remain wary of a Romney candidacy.