Paul Ryan was announced as Mr Romney's choice for vice-president in one such state, Virginia, on Saturday.
The pair are going on to tour North Carolina, Florida, Ohio - and Mr Ryan's home state of Wisconsin.
Mr Ryan vowed to save the country from "high unemployment, declining incomes and crushing debt".
But Mr Obama's campaign has said Mr Ryan stood for what they called "flawed" economic policies that would repeat "catastrophic" mistakes.
Mr Ryan is chairman of the House of Representatives budget committee and authored a controversial alternative budget to counter President Barack Obama's plans in 2011 and 2012.
Profile: Paul Ryan
How media sees choice
Mr Romney, 65, formally unveiled his 42-year-old running mate before hundreds of cheering supporters at the retired battleship USS Wisconsin in Norfolk, Virginia.
In a slip of the tongue, the former Massachusetts governor introduced Mr Ryan as "the next president of the United States", before correcting himself to say he meant vice-president.
"I didn't make a mistake with this guy," he added.
Mr Ryan told the crowd that he and Mr Romney would "restore the greatness of this country".
"Mitt Romney is a leader with the skills, the background and the character that our country needs at a crucial time in its history," Mr Ryan said.
"Following four years of failed leadership, the hopes of our country, which have inspired the world, are growing dim, and they need someone to revive them. Governor Romney is the man for this moment."
Tight race
Prompting one of the loudest cheers from onlookers, he said: "Our rights come from nature and God, not from government."
By Mark MardellNorth America editor
After Mr Ryan was revealed as Mr Romney's running mate in Norfolk, the men stopped in Ashland, Virginia, before embarking on the next step of their tour of swing states, flying from Virginia to North Carolina.
In a little over two weeks' time, Mr Romney will be formally confirmed as the Republican nominee at the party convention in Tampa, Florida.
Mr Ryan chairs the budget committee in the Republican-controlled House of Representatives, and is seen as likely to add electoral firepower on what are expected to be the key election issues - jobs, the economy and the budget deficit.
Recent opinion polls suggest a close race between Mr Romney and President Obama, with Mr Obama tending to have a slight lead in most surveys.
The US Republican presidential candidate, Mitt Romney, and his running mate are touring key states that they need to win in November's election.