Vice President Mondale took his first major campaign trip into Republican territory this week, opening an offensive aimed at winning four states that the Carter-Mondale ticket lost in 1976.

Despite polls showing the president trailing badly in California and Iowa, Mondale stressed the progressive traditions of the two states as the basis for his hopes of victory on Nov. 4.

And in identifying himself as a progressive, Mondale invoked the name of his Minnesota mentor, the late Hubert H. Humphrey as often as he mentioned President Carter.

After Mondale's rousing Democratic National Convention speech last month assailing Republican Ronald Reagan, it appeared that the vice president would be a hatchet man in the campaign, leaving Carter to take the high road. In George Bush's term, Mondale was to be a "groin kicker."

Yet, as Mondale campaigned in Iowa, California, Washington, Oregon and in Wisconsin, one state Carter carried in 1976, he stressed Democratic unity and the party's post-New Deal record. The kicks were few.

Mondale's low-key style made him more nudger than kicker. He jabbed at Reagan and Bush from time to time, mostly in response to reporters' questions, but what kicks he delivered were at less than full force.

Instead, Mondale's first days on what he expects will be a 200,000-mile campaign trip were more phlegmatic than fiery, more in the mode of comfortable front-runner than underdog.

His audiences in Iowa and California generally were like Mondale -- polite and unemotional. A rally of labor officials in Los Angeles and persons in the crowd on San Francisco's Powell Street watching Mondale ride a cable car gave him his most lively receptions.

"We've got to carry California. This is going to be the battleground," Mondale told the labor rally. He urged those in the audience to register new Democrats and get out the vote in an effort to upset Reagan in the state that twice elected Reagan governor.

Carter's California chairman, Mickey Kantor, said he plans to register 350,000 new voters. Carter lost California to Gerald R. Ford by 140,000 votes four years ago and trailed Reagan by 16 percentage points or more in polls taken before the Democratic convention.

Kantor said the strategy this year will be to improve Carter's 1976 showing in areas that often vote heavily Democratic. In San Francisco, for example, Carter won by 30,000 votes, but Gov. Edmund G. (Jerry) Brown Jr. won by 105,000 votes in his 1978 reelection. In Los Angeles, Carter won by 47,000 votes, but Brown beat his opponent by 440,000.

The independent candidacy of John B. Anderson is particularly troubling to Carter's California forces. Polls on a head-to-head Carter-Reagan race show the president trailing, but much closer to his prime opponent than in polls that include Anderson.

"To stay home on election day is to vote for Ronald Reagan. To vote for a third party candidate is to vote for Ronald Reagan," Mondale told the labor rally.

In California, as elsewhere, one of Mondale's key word in unity.

Unity breakfasts, unity luncheons and unity press conferences are scattered through his schedule. At all of them, Mondale welcomes prodigal Democrats who had opposed the president's renomination, and he heaps superlatives on them from a seemingly inexhaustible Mondale word bank of "greatests" and "finests."

Brown, who sought to unseat Carter earlier this year, led Californians rallying to the president and vice president in Los Angeles. Mondale called the California governor a man who "has had and will continue to have a very profound influence on the United States."

"This is a state that will confound and surprise the experts," Brown said in predicting a Carter victory. "I'm here to say that I'll do everything I can to see that that occurs."

Mondale aides denied that their California strategy is more a feint to force Reagan to spend time in the state defending his lead, rather than a genuine effort to win. Kantor said he expects Carter will make three trips to California and Mondale will make four during the campaign.

Aides to the vice presidents said it was too early to predict how much time the candidates will spend here, but the first Carter visit -- a trip with Sen. Edward M. Kennedy -- has been set for Sept. 22.

In Iowa, Democrats base their optimism on a recent poll that shows Carter picking up 10 points on Reagan and their belief that they can make a better case for Carter farm policies than they have so far.

Despite his improvement in the Aug. 21-24 poll, Carter is trailing by 17 points, but Mondale aides argue that Reagan's recent gaffes must have closed the gap further and that they have a chance to win independent and moderate Republican votes. They were encouraged when Republican Gov. Bob Ray, the state's most popular politician, criticized Reagan's stumbling statement on China policy.

The Carter-Mondale strategy depends in part on Anderson's campaign collapsing.

Carter lost Oregon to Ford by only 1,700 votes and Democrats count on environmentally and socially concerned Oregon voters to reject Reagan -- and not turn to Anderson.

In Washington, for similar reasons and because it has a Democratic governor, two Democratic senators and voted for Humphrey in 1968, Mondale aides think that Carter can win.

Mondale's campaigning will be concentrated on the Pacific Coast and the northern states from Wisconsin through New York where traditional Democratic policies and the Humphrey name bring applause.

The vice president will try to explain to Americans turned off by politics that this election presents a sharp choice for the future.

Not everyone who comes to hear Mondale speak apears reachable.

One elderly man turned to his neighbor at a rally in Los Angeles as the vice president was introduced and asked, "Walter what?" Then he took a pamphlet titled "Retirement Plans" from his pocket and wrote down the name "Walter Monday."