Batman’s first case is revealed in Detective 265 (March 1959), in a story by Bill Finger and Dick Sprang.
Batman and robin encounter the Clock, a foe with a with a garish costume and a thing about clocks, and Batman realizes that he recognizes the man from his very first night out in costume.
The story briefly recaps Batman’s origin, and then details his first night out. He stops a thief, but falls for his plea for help. Batman shows a little nervousness in the scene, and thinks about his new rope, which was, indeed, the first piece in his arsenal.
Batman catches up to the man, and captures him. He spends his years in prison brooding over his vengeance, and the concept of time, and clocks.
So when he gets released, he goes after Batman as a costumed villain. Ultimately, he falls prey to the giant gears of the machine he is trying to rob.
Although this story remained canon all the way up to Crisis, the Clock failed to grab the imagination of later writers, and was never really shown again, aside from flashbacks to this story.