For Travis, the definition of adulthood is "being a man." That means many things. Mostly, it means hard work. According to his Papa, a man has many responsibilities. He is expected to take care of his mother and little brother, look after all the farm work, and take initiative to do what needs to be done. A man takes care of all these things stoically, without complaint, and without childish displays of emotion. That is Travis's vision of adult manhood.
As a result of taking on a grown man's work, he begins to see himself as more of an adult. He shakes his father's hand—the first time he'd ever "shaken hands like a man." It made him feel "big and solemn and important"—all manly qualities he believes an adult should have. Also, he extracts the promise of a horse from his father. A man, after all, should have a "man's horse to ride."
Travis's story in Old Yeller is, essentially, a coming-of-age story. He gradually grows from a boy into a man through the experiences he has. Not only due to the responsibilities he takes on but most importantly through his relationship with the dog named Old Yeller. Manhood, Travis learns, is not as glamorous as it originally seems.