Tracy Austin and Pam Shriver, the 16-year-old wunderkinder of women's tennis, took steps yesterday toward a probable quarterfinal showdown in the $125,000 Avon championships of Washington.

Austin -- who turned pro in October and is spending two weeks at a time on the women's tour this winter, moonlighting from her 10th-grade studies at Rolling Hills (Calif.) High School -- easily won her first-round match over Barbara Hallquist, twotime women's national intercollegiate champion from the University of Southern California, 6-2, 6-2.

Shriver, an amateur combining her junior and senior years at the McDonogh School in Baltimore, reached the quarters by running the last 15 points from 3-5 down in the second set to beat Hana Strachonova, 6-2, 7-5.

If Austin wins her second-round match today against Diana Desfor, a Phi Beta Kappa at UCLA who lost to Austin in the first round at Wimbledon last June, she will meet Whriver Friday at George Washington University's Smith Center.

Two seeded players were beaten yesterday. No. 5 Rosemary Casals, still trying to regain her mobility and touch after knee surgery last summer, was ousted by lefthander Ilana Kloss, 6-4, 4-6, 6-3. No. 8 Mima Jausovec fell to Ann Kiyomura, 4-6, 6-4, 6-3.

For a place in the semifinals, Kloss will play unseeded Anne Smith, the athletic 19-year-old Tezan who upset 1977 Wimbledon runner-up Betty Stove in the first round. Smith defeated Janet Newberry, 7-6, 6-2, yesterday.

Also reaching the quarterfinals was Dianne Fromholtz, the No. 4 seed, who shelled Peanut Louie, 6-2 6-0.

Austin, seeded second in the first stop on the 11-tournament $1.9 million women's tour, was cruising along against Hallquist until she doublefauled twice in a row to lose her serve at 4-1 in the second set. She served another "double" to start the eighth game, but won the next four points to close out the match.

Shriver, playing in her first tournament since becoming the youngest finalist in the history of the U.S. Open last September, was lethargic and rather sloppy in the second set until Strachonova served for it. Then she gathered her powerful game and allowed her steadier but less spectacular rival -- who turned 18 Tuesday -- only one more point.

"I just all of a sudden put it all together," said Shriver. "I was kind of mad at myself for gettinginto that position, but I was happy that I came through the way I should at 3-5."

Shriver was not pleased with her approach shots and her failure to sprint for some balls.