The nation's governors yesterday turned their longstanding budget battle with the Reagan administration upside down by proposing that states shed their responsibility for several costly programs, including Medicaid, and take over most federal highway programs and the money to pay for them.

A report, issued as governors began arriving for their annual winter meeting, criticized President Reagan's budget as a "a one-way street" that shifted responsibility to states without giving them the resources to pay for it.

But in issuing the report, Tennessee Gov. Lamar Alexander (R), chairman of the National Governors Association, played down differences with the administration and concentrated on proposals that call for a major reordering of the relationship between federal and state governments.

"I expect we governors will make fewer statements lecturing senators and congressmen on how to do their jobs and more statements on why we can be better chief executives," Alexander said.

One of the report's far-reaching proposals is that the 9-cent per gallon federal gasoline tax be turned over entirely to states, and states assume responsibility for all highway programs, except the interstate highway system. One cent of each nine cents collected is now reserved for mass transit programs. Under the proposal, states would choose how to spend that money.

The report -- "Federalism and the States 1986" -- also proposes that the federal government take over the entire Medicaid program and that the states take greater responsibility for protecting the environment and administrating employment offices, job-training programs and unemployment insurance.

"My own view is Washington ought to work on war and peace issues, welfare, income maintenance, Social Security and reducing the debt," Alexander said. "We the states ought to focus on better schools, clean water, healthy children, roads, prisons and mental health."

Alexander said the proposals are a response to the "new federalism" forced on states during Reagan's presidency and have yet to be endorsed by other governors.

The governors' three-day meeting will include sessions with Reagan, congressional leaders and Cabinet officers. The changing federal-state relationships and educational quality are the top items on the agenda, but the farm crisis, taxes and the impact of the Gramm-Rudman-Hollings budget-balancing law are also to be discussed.

Yesterday, 16 of the governors met at the White House with Secretary of State George P. Shultz, budget director James C. Miller III and other officials to discuss budget and trade issues.

The governors association, dominated by Democrats, supports efforts to reduce the federal deficit, but argued in its new report that cuts in defense spending and tax increases should be part of that effort. The report indicated many governors believe the president is moving unilaterally through the budget process to implement his ideas. A separate study of the president's budget issued by the governors said it would reduce federal aid to states and localities 25 percent below the cost of current services.

The governors repeated their opposition to what they called "attempts to shift current federal costs back to states and localities under the guise of federalism."