Following are some of the major activities on this week's agenda for the White House and Capitol Hills : WHITE HOUSE

Oil, steel and peace in the Middle East dominate the schedules of President Carter and ranking executive aides. Also on tap will be the final legislative program of the adjournament-bound Congress, tax recommendations for the next session of the lawmakers, and a cross-country trip to gauge the direction of the political winds.

Oil, gas and other energy legislation - not too similar to Carter's proposals, but nearing final action in the Senate - will be uppermost on the minds of President Carter's energy and legislative advisers. The President's congressional liaision team will be striving to get the most it can from the Senate as it brings to a close its month-long consideration of the Carter energy packages. ALthough no House-Senate conferences to iron out differences between differing bills ahve been scheduled, Carter aides will be intouch with members of both houses who are to be named conferees. The biggest push will be for energy conservation measures. This was pointed up by Energy Secretary James Schlesinger in Paris when he warned Western nations about the consequences if the world continues to squander oil.

Steel - including foreign imports, sagging U.S. production and factory layoffs - will be the subject of a Cabinet-level executive mansion session Thursday. Carter's special trade representative, Robert Strauss, will chair the meeting, expected to be attended by Labor Secretary Ray Marshall, Commerce Secretary Junaita Kreps and the President. Marshall has a speaking engagement that same day - before the National Press Club - to discuss and then answer questions about Carter's labor policies. The Marshall press club date comes one week after House passage of the labor bill and Senate approval of the minimum wage measure, two big goals of organised labor for which the Democratic-controlled Congress and White House will take political credit.

Carter's political trip beginning next week will be under study by campaign strategists. The five-stage swing will include stops in Michigan, Iowa, Nebraska, Colorado and California. It is expected that Carter will use the trip to plug for issues ranging from the new Panamal Canal treaties to welfare reform. It is to end Oct. 22 with a big fund-raising dinner in Los Angeles.

The Middle East peace efforts that began with White House conferences with Israel and Arab foreign ministers, then moved to the United Nations, will resume this week in Washington at the ambassadorial level. One of the principals - Egyptian Foreign Minister Ismail Fahmi - will be passing through Washington, ostensibly to catch a concorde for Paris. He could pay a call on either Carter or Cyrus Vance before he boards his airplane.

SALT, Africa, as well as other international areas, also will be on various executive mansion desks. A new SALT pact, which Carter says is "within sight," will be high on the Vance calendar. Africa will be emphasized in the official visit Tuesday and Wednesday by Nigeria's chief of state, Lt. Gen. Olusegun Obasanjo. Panama, Cuba, and China are other nations that will continue to get close White House attention. China is among the Asian nations that Carter hopes can establish full diplomatic relations with the United States "without delay."

What can be expected from Congress for the few weeks left of the current session will be taken up at both Cabinet and congressional leadership sessions this week. Revised tax proposals that will be sujitted to Congress early next year, the new welfare program, and anticrime measure probably will be among topics at these meetings. Carter spent the weekend at Camp David, with the new tax proposals as his homework. CAPITOL HILL

The House meets four days this week, with nine bills on its calendar. The Senate has scheduled a full week, but it's anybody's guess when the energy tax bill, the final piece of the Carter energy package that has not been formally reported by one or both chambers. Today is Columbus Day and the House is taking the day off. The Senate, with is calendar bulging as a result of the natural gas filibuster, will meet today through Friday, and possibly on Saturday.

Two bills are scheduled for Senate consideration today. The legal services measure will be considered first. Then the Senate, at 3 p.m., will call up a bill to clamp down on child pornography. The legal services bill will then be called up again, if action has not been completed.

The Senate Finance Committee is resuming work on the energy tax bill on Tuesday. It is hoped that the legislation will be ready for the floor no later than the end of the week.

In the House, two bills have been set for debate on Tuesday - The ocena dumping measure and the navigation development act. Votes on both, however, willbe put over until the end of the week. The compromise foreign aid appropriations bill will be called up Wednesday. On Thursday and Friday. the House is to take another vote on the abortion-bogged Labor-HEW compromise; act on the supplemental appropriations bill, FTC amendments, government organization plan No. 1 (executive offices), and the federal banking agency audit act.

Congressional committees, despite the few remaining weeks of the current session, continue to grind out legislation and sponsor hearings on a host of issues. Hearing on the Panama Canal treaties are scheduled to end, with Secretary of State Vance appearing before the House Intertional Relations Committee and former Secretaries Henry A. Kissinger and Dean Rusk testifying befor the companion Senate committee on Friday.

The Senate Finance Committee is believed ready to end its markup of an energy tax bill, so chopped up that many consider it worthless. Nezt on the Finance Committee's schedule are hearings on Medicare and Medicaid revision, starting Wednesday. The House Ways and Means Committee, meanwhile, will continue on welfare revision.

Legislation to authorize a pipeline to bring natural gas from Alaska will be before both House and Senate committees. A House Interior subcommittee will hold hearings on a coal slurry pipeline.

The Senate Judiciary Committee meets Tuesday on the nomination of Frank Johnson to be the new director of the FBI. Nominations to the Corporation for Public Broadcasting will be considered by the Senate Commerce Committee, also on Tuesday.