For his part, Mr. Bush said he was glad to have met Schwarzenegger, then joked about "how much we have in common."
Both "married well," said Mr. Bush, and "some accuse both of us of not being able to speak the language."
Mr. Bush joked that he and Schwarzenegger also had a third thing in common: "big biceps." When the laughter subsided, Mr. Bush suggested that "two out of three isn't bad."
The president and governor-elect met privately in Mr. Bush's hotel suite on Thursday in Riverside before they rode together in the president's limousine to nearby San Bernardino, where Mr. Bush gave a speech designed to set the stage for his Asia trip, which will focus on trade and the war on terror.
Both men drew tumultuous applause as they were introduced at an economic forum.
"Thirty-five years ago, I never imagined I would be standing in front of you as governor of California introducing the president of the United States," Schwarzenegger said.
California, Schwarzenegger said, is "a place where dreams come true" — though its staggering budget deficit made for "serious challenges."
The president said he was "confident about the future of our economy," and noted that he had the responsibility to govern wisely from Washington while "Governor Schwarzenegger has the responsibility to set good policy in Sacramento."
On other issues, Mr. Bush said he welcomed Thursday's unanimous vote by the U.N. Security Council aimed at attracting more money and troops to stabilize Iraq — a big diplomatic victory for the administration after a string of recent setbacks.
"I want to thank the United Nations Security Council for unanimously passing" the resolution, he said.
"America must never forget the lessons of Sept. 11," Mr. Bush said. "America will never retreat...We will fight this war against terror until it is won."
Mr. Bush also netted some $1.75 million for his re-election campaign with a fund-raising luncheon in Fresno and a dinner fund-raiser here, bringing his overall campaign war chest to roughly $84 million.
At the private meeting between Mr. Bush and Schwarzenegger, lasting about half an hour, the two talked about their "shared views of governance and how to approach issues," said White House communications director Dan Bartlett. He said the meeting "set a positive tone for a good working relationship going forward."
"It was a good, substantive conversation," Mr. Bush said.
As they waited just offstage to be introduced, Mr. Bush and Schwarzenegger were overheard joking about how neither one had a good grip on pronouncing the California city Rancho Cucamonga.
White House political strategists hope the Schwarzenegger era will force Democrats to defend the big electoral vote prize that Mr. Bush lost by a million votes in 2000, reports CBS News Correspondent Peter Maer.
But House Democratic Leader Nancy Pelosi of California suggests the same line Republicans successfully used against Gov. Gray Davis could be turned on President Bush in the coming campaign.
"He inherited record surpluses, he turned them into record deficits. I think President Bush should take heed of that pitch," Pelosi said.
And conservatives in the president's own party say he should take heed of their concerns about Schwarzenegger's support of abortion rights and gay rights. Distancing itself from the social issue questions, the White House says the president and the actor-turned-politician will find shared priorities.
Mr. Bush's speech in San Bernardino was his last event in California before leaving on a six-country trip to Asia and Australia, stopping first in Japan.
He got some good news ahead of his visit to Tokyo — Japan's pledge for $1.5 billion in the coming year for reconstruction projects in Iraq, and possibly as much as $5 billion over the next four years.
Mr. Bush said Japan's commitment would provide aid for immediate reconstruction needs there.
"I applaud this bold step which will help mobilize international support for efforts to build a stable, peaceful and democratic Iraq," he said. "Japan recognizes that this effort is critical to security and peace not only in Iraq and the Middle East, but also for Japan and throughout the world."