Robert M. Gants announced last week that he will seek the Republican nomination for Alexandria City Council next spring, saying he wants to hold down the city's tax rate but hold on to its high quality of life.

All six members of City Council and the mayor will be elected in citywide balloting May 10. The Republican Party will choose its candidates in a party canvass, which is similar to a primary election, in February or March. The date will be set early next month.

Gants, 50, a Capitol Hill lobbyist for the small-appliance industry and an unsuccessful candidate for the Virginia House of Delegates this fall, told officials of the Alexandria Republican Party that he wants to "complete the job" he began in this year's campaign.

"We cannot raise taxes. Just throwing money at a problem is no way to solve it," Gants said. "I bring a steady hand, more than a little political savvy and a willingness to listen to all sides as we seek and develop imaginative new ways to deal with stubborn problems."

Gants, an Alexandria resident for 18 years, is a graduate of West Point and the American University School of Law. He worked as a staff lawyer and top aide to several Republican congressmen in the 1970s before becoming a lobbyist. He also founded a civic association in his neighborhood. Cleveland Cites Growth as Priority

William C. Cleveland announced last week that he will seek the Republican nomination for Alexandria City Council next spring and said that his priorities will include controlling growth and increasing police protection.

All six members of City Council and the mayor will be elected in citywide balloting May 10.

Cleveland, 39, who has worked for 14 years as a U.S. Capitol Police officer, narrowly missed being elected to the council in 1985, finishing seventh in the contest for six seats. He was less than 200 votes behind the sixth-place finisher.

"We have to slow down growth in Alexandria, and we have to do something about lowering the zoning density for commercial structures," Cleveland said. "We also need more protection in our city. Our police are stretched too thin."

Cleveland moved to Alexandria after serving three years in the Army, including a tour of duty in Vietnam. He set up the city's first Neighborhood Watch program in the Warwick Village area where he lives. According to Cleveland, the number of burglaries in Warwick Village dropped from 47 in 1979 to four in 1982. "I think I can provide leadership," he said. Clinic Prompts Kite To Enter City Race Saying that he opposes both a proposed clinic that would dispense contraceptives to Alexandria students and a sex education course taught in city schools, Richard L. Kite announced last week that he would seek the Republican nomination for City Council next spring.

All six members of City Council and the mayor will be elected in citywide balloting May 10.

Kite, 45, an electrical engineer with American Amplifier and Television Corp., is making his first bid for elective office. He has lived in Alexandria since 1973 and has been active in local GOP politics since 1980, doing precinct work for a variety of candidates.

"The motivating factor for me to throw my hat in the ring was the clinic," Kite said. "I think the council needs some more representative opinions on the subject. Most of the people who attended a public hearing on clinic were against it, but a majority of council is for it.

"In the past I've been a supporter of candidates, and I think I've done a pretty fair job. Now I want to find out what it takes to win an election on my own." Williams Promises Tax Cut if Elected

Charging that Alexandria's city government is inefficient and promising to cut local taxes, John D. Williams III announced last week that he will seek the Republican nomination for Alexandria City Council next spring.

Williams, 49, a real estate broker, has run unsuccesfully for council twice and for mayor once in the past three years. He is president of the Alexandria Taxpayers Alliance, an antitax group, and is a familiar face at council meetings, where he frequently urges city officials to reduce taxes and the municipal work force.

"Our taxes and spending have gone up three times faster than inflation for the last 20 years," Williams said. "It's outrageous. We can easily lower taxes by curtailing the rapid growth of the City Hall crowd."

Williams graduated from the U.S. Naval Academy in 1960 and served in the Navy for 10 years. He managed his own computer firm from 1970 until this year, when he entered the real estate business.

Williams has long been active in antitax organizations and is a vocal opponent of a proposed teen-agers' clinic that would give health checkups and dispense contraceptives to students at Alexandria's only high school. Caprara Opposes School-Based Clinic Calling a proposed teen-agers' clinic that would give health checkups and dispense contraceptives to Alexandria high school students "a grave and immediate concern," David Caprara announced last week that he will seek the Republican nomination for Alexandria City Council next spring.

Caprara, 32, a policy analyst who specializes in urban affairs for the National Center for Neighborhood Enterprise, has never before sought elective office. He has been active in a group that opposes the clinic, Concerned Alexandrians for Responsible Education, and has previously served as an aide to Sen. Alan Dixon (D-Ill.)

"The city's blind rush to enact a contraceptive-based clinic would undermine parental authority," Caprara said. "We should instead adopt an aggressive campaign designed to prevent teen-age pregnancy." Caprara has previously helped organize tenants in low-income housing projects, including the troubled Cabrini Green Homes in Chicago, and says that Alexandria should do more for its poor. "I would like to empower the residents of public housing, give them more control over their lives," he said. Clinic Fight Puts Mulloy in City Race

Marjorie Mulloy, the president of a group fighting a proposed teen-agers clinic that would do health checkups and dispense contraceptives to Alexandria high school students, announced this week that she will seek the Democratic nomination for Alexandria City Council this spring.

All six members of City Council and the mayor will be selected in citywide balloting May 10. The Democratic Party will select its candidates in a primary election tentatively set for March 8. Mulloy, 42, a special education teacher in the Fairfax County school system, has never sought elective office. She became president of Concerned Alexandrians For Responsible Education when the clinic proposal came before city council this fall. She has a doctorate in early childhood education from the University of Maryland.

"I just feel like the council members and the city officials haven't been responsive to a large portion of the city's population," Mulloy said. "I'd like to get in and change that."

Mulloy also said that she would like the city to slow its growth, which she said "is getting out of hand," and offer more day-care services to teen-age mothers. "I am not a one-issue candidate," she said. "I'm interested in all aspects of Alexandria's life."