In the first Republican presidential cattle show for the 1988 elections, Vice President Bush and Rep. Jack Kemp (R-N.Y.) gave a preview of the battles that lie ahead to inherit the political legacy of President Reagan.

The early stirrings of the public battle for the GOP nomination were prompted by the gathering of 500 party leaders from 14 states at the Midwest Republican Leaders Conference, many of whom will be critical players in their states' caucuces and primaries.

Bush, Kemp, Senate Majority Leader Robert J. Dole (R-Kan.) and former Delaware governor Pierre du Pont also were drawn to Michigan because the state's GOP has adopted a presidential-nomination process that will start in an August 1986 election, two years ahead of the 1988 presidential convention, and a year and a half before the Iowa caucuses.

Approximately 10,000 "precinct delegates" will be selected around the state next year. These delegates will form the base, through a series of caucuses in 1988, from which the Michigan delegates to the 1988 national GOP convention will be elected.

Unless the Michigan GOP changes the rules, candidates seriously contesting for Michigan's delegates in 1988 will have to be prepared to field full, or close to full, slates of 10,000 candidates for precinct delegate in August 1986.

All four of the prospective candidates arrived here armed with their political staffs and political action committees (PACs) flush with millions of dollars in funds.

Bush was in town only briefly on Saturday night, but Ron Kaufman, a key political operative for Bush and director of the vice president's PAC, the Fund for America's Future, stayed through today to line up support in this state, which backed Bush over Reagan in 1980.

Kemp not only brought John Maxwell, director of his PAC, Campaign for Prosperity, but Roger Stone, a campaign consultant who has specialized in the Northest and Midwest. In an effort to add credibility to the Kemp drive, supporters said they expect pollster Lance Tarrance to join them soon, countering the presence of pollster Robert Teeter in the Bush campaign.

Bush and Kemp avoided head-on confrontations, as Kemp chose not to attend the Saturday night dinner at which Bush was keynote speaker.

Kemp's absence did not, however, prevent his backers from passing out invitations to all those attending the Bush dinner to a special hospitality suite afterward.

Bush held a more exclusive, private gathering after the dinner in a 26th floor suite of the Amway Grand Plaza Hotel, owned by the Amway Corp., the chief officers of which are major Republican fund-raisers and, according to Bush aides, backers of the vice president.

"We'll have the leadership in this state," a Bush strategist said. "Kemp's people are going to claim they have the grass roots while saying we are going after the establishment," he added.

"I will not be hyping anything," Bush told a crowded news conference before his speech. "I believe that the best politics is to do my job."

"Our basic strategy is simple: Bush will be presidential," an aide to the vice president said.

John Buckley, Kemp's press assistant, contended, "Jack is putting together a campaign of ideas, not a campaign of endorsements. This will not be an incrementalist Mondale campaign of endorsements," in effect likening the Bush camapign to that of Democratic presidential nominee Walter F. Mondale in 1984.

State Senate Majority Leader John M. Engler is coordinating the Bush drive. W. Clark Durant III, a Detroit lawyer recently approved as chairman of the Legal Services Board, has formed the Michigan Opportunity Society, which is functioning as Kemp's local political organization.

While most attention focused on Bush and Kemp, Dole appealed to traditional Republican concerns about the deficit and called for enactment of a balanced budget constitutional amendment, while du Pont called for consideration of a "post-Reagan agenda." Sounding somewhat like Sen. Gary Hart (D-Colo.), du Pont, who was well received at the gathering, said the GOP "can't be a party that says more of the same or less of the same."