When President-elect Carter began his inauguration day by attending a private service at First Baptist Church, and agreed to attend there again this Sunday, he put the church on 16th Street NW. ahead in the spirited competition for the new President's congregational affiliation here.

Dr. Charles A. Trentham, the wavy-haired, courtly pastor of First Baptist, who has been lobbying since last spring for Carter's presence at his church, beamed as he greeted Carter yesterday morning and made the most of the occassion.

"You are now in a church which has pledged to pray daily for you," Dr. Trentham told Carter in front of members of his incoming Cabinet, high-ranking White House staff and their families after they were seated.

That pledge was made last Sunday, he said.

"I believe in Jimmy Carter," he went on. "I voted for him.On the day he was nominated, I thanked God for the mergence of a new leader and convenanted to daily call his name and the name of each member of his family in prayer."

President-elect Carter said during the campaign he would worship at the Southern Baptist church closest to the White House, but he later learned two churches are equidistant form the Executive Mansion. The 175-year-old First Baptist church is six blocks north on 16th Street at O Street NW and Calvary Baptist is six blocks east at 8th and H Streets NW.

Asked yesterday if First Baptist was Carter's final choice, as his place of worship, Dr. Trentham said with a smile from ear to ear."That is not my announcement to make."

Over at Calvary Baptist last week, Dr. George Hill, the pastor, noted that Carter had written him Sept. 11: "I do hope to have the opportunity to visit with you." Dr. Hill had invited Carter by letter.

"I consider that a promise, that he will visit our church before he makes up his mind," Hill said at the time. Meanwhile, Baptists here are speculating that other more distant churches are also in contention.

Dr. Stuart L. Grizzard, pastor of National Baptist Memorial Church at 16th Street and Columbia Road NW extended an invitation to Carter in a letter after he had met Carter at a meeting here last May.

"Back then," Dr. Grizzard claimed, "his Washington organization was advising him to this church when he was in town."

Still another possibility being discussed is the small Riverside Baptist Church in Southwest where Rep. John Buchanan (R-Ala.), a Southern Baptist minister, belongs.

Spokespersons for the Carter family insist no decision has been made. A Carter staff member is known to have prepared a detailed memo for the President about how other chief executives worshipped in Washington and about the characteristics of the various churches. Chip Carter has quietly visited both First Baptist and Calvary Baptist since his father's election.

Carter requested yesterday's private worship service several weeks ago and did not attend the public interfaith ceremony at Lincoln Memorial.

Joan Mondale's father, the Rev. Dr. John Maxwell Adams of St. Paul, Minn., conducted the 45-minute ceremony, and the Rev. Nelson L. Price, pastor of Roswell Street Baptist Church in Marietta, Ga., who has been Carter's daily "prayer" partner" for eight years, gave a brief sermon.