Though the title is Lady and the Tramp: A Trusty Old Pal, the two main characters of the story are Lady and Tramp's friend, Trusty, and son, Scamp.
Trusty likes to reminisce with the puppies, but he has the tendency to repeat himself. And after one too many times hearing the same story, Scamp can't take it anymore. He finally tells Trusty what he thinks of that story, and Trusty's feelings are obviously hurt. Though he was supposed to be staying with the family, Trusty's gone by morning.
So the family splits up to look for their lost friend; the females take one half of town and the males take the other. As one can guess, Scamp ends up in danger during the search, and who should save him but Trusty. Because there could be no other outcome, Trusty and Scamp patch up their relationship, and Scamp and the other puppies end up with a newfound appreciation for their oldest friend.
All in all, it's not a bad story, but it's not exceptionally interesting either. There's an obvious moral of respect and sparing other's feelings in there, but there's also a subtle suggestion of suffering in order to spare other people's feelings that I'd certainly question. In any case, it's an acceptable little story; the animation's nice, it's fun to see Disney characters getting up to things post-movie (as long as it doesn't, you know, shit on continuity), and this installment has none of the moral!fail of some of the others in the series.