Sid Caesar; native American rights; and more; Caesar's reign as comedy king; Where Have I Been, by Sid Caesar with Bill Davidson. New York: Crown Publishers Inc. 280 pp. $12.95.

It's a familiar ''show biz'' story. Success that comes too easily, too soon.

In the 1940s, Sid Caesar earned six dollars a week playing saxophone in a dance band; within a few short years, he commanded $25,000 per TV show as the king of comedy. Then, inevitably, his program was cancelled. He passed through the '60s in an alcoholic stupor; the '70s found Caesar muffing his lines on an obscure Canadian stage.

In this case, however, the story has a happy ending. After years of unsuccessful psychoanalysis, Caesar succeeded in accomplishing from within what no one could do from the outside. He climbed on the wagon, learned to control his rages, and patched things up with his wife and children. He began cultivating flowers, and picked up the pieces of his career.

With the help of coauthor Bill Davidson, Caesar's autobiography races along in 30 short chapters - engrossing and straightforward, if sometimes a bit slick. There is plenty of reminiscing about the ''Your Show of Shows'' days, including anecdotes about Carl Reiner, Mel Brooks, and others who helped make it a classic.

While occasionally amusing, this is not a book written for laughs. Caesar states that his main purpose is to give hope to readers suffering from tough personal problems. He writes, ''. . . if I could learn to conquer my overwhelming fears, addictions, and self-doubts, certainly others can learn to banish their own demons.''