He also reiterated his call for tax cuts for the middle class.
Unlike in his State of the State speech last week, Christie made no mention of the multiple investigations of his administration.
His speech came less than an hour after Democratic state lawmakers announced they would merge investigative committees from the Assembly and Senate into one panel to look into lane closures on an approach to the George Washington Bridge.
Emails released this month showed a top Christie aide ordered the closure of approach lanes in the town of Fort Lee apparently as political retribution against the mayor there. And new questions have arisen about his use of post-Superstorm Sandy recovery aide.
"We have to be willing to play outside the red and blue boxes," Christie said. "We have to be willing to reach out to others who look or speak differently than us."
Christie also returned to a favorite theme: Washington gridlock.
"We cannot fall victim to the attitude of Washington, D.C.," he said. "The attitude that puts political wins ahead of policy agreements."
The 55th governor of New Jersey had to modify the schedule of inaugural events because a severe winter storm threatens to dump up to a foot of snow on the region Tuesday.
His day started with a service at Newark's New Hope Baptist Church before the swearing-in and address in Trenton.
But organizers canceled an evening party on Ellis Island, a symbolic spot synonymous with the promise of the United States, because there was fear snow would make travel dangerous. The island where some 12 million immigrants first entered the U.S. is divided between New Jersey and New York, but his party was supposed to be held in a hall on the New York side.
Food prepared for the party will instead be donated to food pantries in the Jersey City area.
Lt. Gov. Kim Guadagno, who was drawn into the Sandy aid controversy surrounding Christie last weekend, was also sworn in for her second term.
Christie won re-election in November by a 22-point margin over state Sen. Barbara Buono, a Democrat.
The Republican governor built a national following as a blunt-talking and often funny politician who strived to show that he could find common ground with Democrats on some key issues, including overhauling the state's public-worker pension program and making it easier to fire teachers who are found to be underperforming.
Christie became a fixture in speculation about who would seek the 2016 presidential nomination with his leadership after Sandy slammed into his state in October 2012.
He worked with President Barack Obama and took on Republican members of Congress who were reluctant to approve aid for storm victims, receiving high marks from his constituents and plentiful national attention, along with criticism from some conservatives.