The Senate yesterday overwhelmingly rejected a plan to force drivers to leave their cars home one day each week, and then went on to approve a measure telling the president and the governors to devise their own energy-saving plans.

It was the first floor action on energy conservation since the House killed President Carter's standby gasoline rationing plan last month, and the legislation approved was drafted as an alternative to that plan and other conservation measures rejected earlier by Congress. It now goes to the House.

The Senate action was also the first test of sentiment on asking drivers not to use their cars once a week, but because of the way the amendment was drafted, the 79-to-10 vote rejecting it may not be a true indication of congressional feeling.

The amendment, introduced by Sen. Lowell P. Weicker (R-Conn.), called for a mandatory program within 60 days after enactment of the legislation, requiring all cars to bear windshield stickers showing the day of the week they couldn't be driven.

Some senators said the plan was too inflexible, and that may have contributed to the big vote against it.

A less stringent plan, calling for drivers to leave their cars at home once a week only after other conservation measures have failed, has been gathering some support in the House. It is sponsored by Rep. Toby Moffett (D-Conn.)

In another vote yesterday, the Senate gave Carter authority to prohibit hoarding of gasoline. The amendment came after newspaper articles about people buying tanks to store large amounts of gasoline under their yards.

The Senate's energy conservation bill, approved by a vote of 77 to 13, would forbid the federal government, but not the states, to include weekend gasoline station closings in their plans. It encourages states to draft energy savings plans best suited to their region and to send them to Washington for approval. It also would direct the president to draw up a national conservation plan which, in an emergency, would be put into effect in states that had not prepared or had approved their own plans.

When an energy emergency developed, the president would announce a national target for energy-saving, and the state or federal plans would be put into effect. They might include odd-even gas-purchase days, forbid small gas sales, keep the car in the garage one day a week or lower the speed limit.