The 97th Congress, dominated by conservatives and divided by party for the first time in a half-century, convened yesterday with bipartisan vows to cooperate with the incoming Reagan administration.

In the Senate, under Republican control for the first time in 26 years, Majority Leader Howard H. Baker Jr. (R-Tenn.) said he expects that committee action on President-elect Ronald Reagan's Cabinet nominees will be completed by Reagan's inauguration Jan. 20.

And Senate Minority Leader Robert C. Byrd (D-W.Va.) said he anticipates no organized Democratic opposition to the confirmation of Alexander M. Haig Jr. as secretary of state, unless unforeseen problems arise. "If there are not any particular problems," said Byrd, "our role will be to expedite the confirmation process."

In the House, where the Democrats retain control by a diminished margin, Speaker Thomas P. O'Neill Jr. (D-Mass.) promised "constructive dialogue rather than partisan recrimination" in dealing with the Republican White House and Senate. As for Reagan, O'Neill said, "We seek to work with him to meet the challenges we face as a nation. . . . I personally wish him well."

But there was a limit to the generosity of the beleaguered House Democrats, who had the votes to squelch an opening-day bid by Republicans for expanded representation on the powerful Rules Committee and Ways and Means Committee.

The Senate got off to a less rancorous start, but Byrd took advantage of a couple of opportunities to needle the new Republican leaders on parliamentary proprieties -- a possible omen of partisan warfare to come.

Aside from the partisan posturing, it was largely a day for pomp, ceremony, greeting of old friends and swearing-in of new members -- a bumper crop of 18 in the Senate and 73 in the House.

The new Senate has 53 Republicans, 46 Democrats and one independent -- a gain of 12 for the GOP from last November's elections and the party's biggest representation in 50 years. In the new House, there are 243 Democrats and 192 Republicans -- an increase of 33 seats for the GOP, which can also count on 40 or more conservative Democrats to vote with the Republicans on many major programs.

Not since the early 1930s -- when the Republicans held the Senate and the Democrats the House -- have the two houses of Congress been controlled by different parties.

Shortly after Congress convened at noon, senators who won last November -- 16 incumbents as well as the 18 newcomers -- were led to the well of the chamber in groups of four to be sworn in by outgoing Vice President Mondale and applauded by the packed galleries. So numerous were the newcomers that Deputy Minority Leader Alan Cranston (D-Calif.) kept glancing periodically at a reference sheet on his desk that matched the new members' names and faces.

In the House, where all 435 members were sworn in en masse, the ceremony was less stately if no less colorful. As usual, members brought in their small children for the occasion, and the aisles were filled with cavorting youngsters as well as a continuing undertone of conversation that drowned out the official business.

O'Neill was reelected speaker for a third time by a predictable party-line vote of 234 to 183. Once O'Neill had been elected speaker, he was sworn in by the dean of the House, Jamie Whitten (D-Miss.), and O'Neill then gave the oath to all other members. No sooner had O'Neill offered his cooperation to the Republicans than the Democratic majority ran right over the Republican House minority by rejecting, 220 to 180, its request for more seats on Rules and Ways and Means, two of the House's most important committees.

Now that they compose 44 percent of the House, Republicans said the tax-writing Ways and Means Committee should revert from 2-to-1 Democratic to the traditional 3-to-2 ratio of control by the majority. Democrats refused to go beyond 23 to 12 on Ways and Means and refused to give away any of their 11-to-5 majority on the Rules Committee, which schedules legislation for House action.

Democrats also rejected a Republican proposal to write into House rules a requirement that federal spending be held to a specific percentage of the gross national product, which is the total of goods and services produced in the country. Rep. James R. Jones (D-Okla.), new chairman of the Budget Committee, who supported such a plan in the last Congress, said he was committed to getting a vote on the spending limit but said it would be "premature" to write it into House rules now.

In the Senate, Baker introduced a resolution calling for televised coverage of Senate proceedings, calling it "simply a modern-day extension of the public gallery and the public's right to view the legislative process of the government on a firsthand basis." Baker said he believes the Senate will join the House in permitting television coverage. Baker said it will be up to the Rules Committee, to which the resolution was referred, to decide whether the Senate or the television networks should handle the coverage.

Baker also reiterated that Sen. Paul Laxalt (R-Nev.), who was Reagan's campaign manager, will have a role in Senate Republican leadership, including attending congressional leaders' meetings at the White House.