Travis comes across as an incredibly mature young man for his age. It's a good thing that he is because otherwise his father wouldn't have left him in charge of the family ranch while he went off on a cattle drive. Life was hard out there in the wilderness, and you had to be self-reliant and strong if you were going to survive. But fortunately, Travis has both of these characteristics in abundance. By temperament and upbringing he's able to defend his family against the many dangers that threaten them. As well as considerable maturity, his experiences also give young Travis a realistic, unsentimental view of life. Despite the very close bond he's developed with Old Yeller, he doesn't hesitate to shoot him dead when he gets bitten by a rabid wolf. When push comes to shove, Travis instinctively knows that the welfare of his family is more important than a dog, however emotionally attached he is to it.
The first and most remarkable character trait of Travis is his trustworthiness. His father feels comfortable leaving him in charge of the work of the farm and the homestead, and Travis succeeds admirably in being able to take care of everything entrusted to him.
Another might be his quick ability to adapt according to the situation, sometimes for good sometimes not. If you look at his reaction to Old Yeller's death, he is at first unwilling to bond with a new animal, unwilling to form that attachment only to have it broken again. But he is also willing to listen to his father and to try to "participate in life" rather than simply observing it.
One other trait you could point to is his willingness to work hard. You might just contend that it was a necessity for boys at the time, but Travis appears to be particularly willing to put incredible efforts into his work and takes great pride in it.