In the opening of one episode, Barney is upset because he thinks no one remembers his fifth-year anniversary as deputy. Then it turns out that they were all just pretending not to because they had a surprise party planned, complete with an engraved watch. The look on Barney's face when he realizes that they remembered him after all is just incredibly heartwarming.
In "The Cow Thief", when Andy tries to explain to the mayor his theory about how the cows were stolen (the thief fitted them with human shoes to confuse the investigators), the mayor and the state agent roundly insult Andy for wasting their time, calls his idea "hare-brained", and walk out on him. Barney, who had been hero-worshiping the state investigator all week, at first follows them, but a moment later returns to Andy to help him, confessing that he was reminded of another old "hare-brained idea" of Andy's — the idea to make Barney his deputy.
In "Man in a Hurry", the titular man, who has been grousing about being stranded in Mayberry all episode, is nearly moved to tears when he stops worrying and obsessing about work and being late long enough to notice Andy and Barney strumming a guitar and humming "The Little Brown Church in the Vale" while lazing in the sun on the porch, realizing the value of the little things he's been missing in his hurry.
The scene in "Mr. McBeevee", when after a very tense scene where Andy tries to make Opie admit Mr. McBeevee is just make-believe, with the threat of a whipping very clear, Opie still insists that Mr. McBeevee is real, desperately asking "Don't you believe me, Pa?" After a very long silence, Andy gets up, pats Opie on the shoulder, and says, quietly, with complete sincerity, "Yeah, I believe you". He has no evidence apart from Opie's word to support his belief, but according to his explanation afterwards:
Andy: I guess it's a time like this, when you're asked to believe something that just don't seem possible, that's the moment that decides whether you got faith in somebody or not.
Barney: Yeah, but how can you explain it all?
Andy: I can't.
Barney: But you do believe in Mr. McBeevee?
Andy: No...no...no. I do believe in Opie.
In "Opie's Job", after Opie tearfully explains to Andy that he had to purposely get himself fired from his grocer job so that the other trainee, whose father is poor and needed his son to work to help pay the bills, could land it instead of Opie, Andy quietly gives Opie possibly the greatest praise a parent can give their child:
Andy: You know, when I was bragging on you to Floyd and Goober, I told 'em how proud I was to have a boy like you. But that's not quite true. You're a man.
The end of "Opie the Birdman".
Opie: The cage sure looks awful empty, don't it, Pa?
Andy: Yes, son, it sure does. But don't the trees seem nice and full?