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.