James and Sarah Brady have an ambitious goal -- getting him home to stay before Thanksgiving.

Brady, President Reagan's press secretary and the amiable favorite of all who know him, has been a patient at George Washington University Hospital since he was shot in the head and permanently disabled last March 30, when the president also was wounded. Brady is still hurt physically and mentally but determined and recovering.

"In the last four weeks," his wife, Sarah, reported yesterday, "he has made tremendous progress. Everything seems to be right on track at last."

"At last" covers a lot of ground. It covers six hours of emergency surgery on March 30, three more operations, leakages of brain fluid through his sinus passages and nose, blood clots, pneumonia and an epilepsy-like seizure.

"Every time he was making progress, something would happen to put him down," said Mrs. Brady.

But for more than a month now he has been working hard again at four hours a day of physical and occupational therapy, "occupational" meaning relearning daily living. Walking. Standing up. Sitting down. Dressing. Bathing himself.

For the bullet that tore through much of his brain left a permanent legacy of disability.

His left hand and arm remain mainly paralyzed, though it is still possible some use will return. Movement in his left leg is impaired, though he can walk with some help -- a short "elbow crutch" in his right hand and someone at his left side to steady him.

His walking is improving and he should be able to walk alone before much longer, his surgeon, Dr. Arthur Kobrine, reports.

What is really striking, said Mrs. Brady, is that "each time he's come home, he's improved so markedly from one day to the next. It's been a big change in the last month or two."

This also applies to his brain functions.

"He is doing very well mentally," a pleased Kobrine said in a recent interview. "His intelligence, judgment, insight, wit and sense of humor all seem normal." There are a few problems --"occasional problems with recent memory" and "he kind of gets choked up emotionally sometimes . . . . You and I suppress this . . . . He has to relearn that."

But, Kobrine added: "All these things are improving. In the long run I don't expect any bad deficits, only things that are manageable. Because he's using so much energy to recover, for example, his brain sometimes has trouble storing new information. Once the brain is really recovered, I think it will do so pretty normally . . . .

"I can tell you his political comments and insights are superior. I see nothing now to rule out his return to an intellectual white-collar job."

"I don't think there's any doubt in his mind," said Mrs. Brady, "that he'll get himself back to work and an active life."

All this is not easy. "He's had his down time," she said. "He's had to learn to be a very patient person. He's never been very good at sitting and waiting."

And the first time he came home for a full day-- which he first did for three consecutive days over the Labor Day weekend -- "I was so exhausted physically and emotionally," she admitted.

There still could be serious medical setbacks in any such brain injury.

But the main immediate task, she said, is some remodeling of their three-story home and hiring some permanent help, either a daytime nurse's aide or round-the-clock aid. Until some of this is done, in fact, Brady may not be able to spend a full night at home. So far he has continued to go home only for weekend days.

The remodeling would mainly include installing chair-lifts between floors, and wheelchair ramps. According to one source, the remodeling plus help for perhaps a year might cost as much as $100,000.

"I'm still looking at what may be needed before setting any dollar value. Each time he comes home the problems are less," so the needs may be less, she said.

She "understands," she added, that some benefactors have said they will contribute these installations and nursing care. Brady's White House salary, meanwhile, goes on, and his medical and hospital bill will be almost entirely covered by his insurance.

One emotion Brady has not shown, she said, is any bitterness.

One day, by arrangement, he talked on the phone to a boy who had been badly injured. He told him: "We're lucky to be where we are and getting well and getting good care."