Season 1 Episode 19: Coming of Age
Short Answer: A couple interesting points, but nothing you are going to miss.
Notables: Serves as a sort of precursor to "Conspiracy"
There are two main threads running through this episode. The first one is about one of Picard's old friends who is now an admiral. He comes to the Enterprise and investigates Picard's record for reasons unknown. By today's standards has a fairly predictable twist. The more "interesting" part of this plot line is continued in the episode "Conspiracy" later this season, so I won't go into it here. We'll see how I feel when I get to it, but I recall that episode being one of the strangest episodes in the entire series. If you watch this one, definitely watch "Conspiracy". It's not good, but it will probably be good for a laugh.
The second plot line follows Wesley Crusher as he applies to Starfleet. It's interesting for a couple reasons. The one that piqued my interest is that the Starfleet application process is very selective. Of all the children on the Enterprise, Wesley is the only one allowed to take the exam. The plot diverges a bit into the story about another teenager who does not get accepted. I suppose he is intended to be a foil for Wesley, but I have a hard time seeing the purpose. He seems to exist solely to let Picard give him a heart filled pep-talk about trying again.
Having the exam be so selective definitely seems to drive home the point that the Starfleet is "the best and the brightest" (which later plot-lines are happy to contradict). We already saw that it is difficult to even be allowed to take the test. In addition, only one person from the test is accepted. It's understandable that Starfleet wants people of a certain caliber, but having the applicants compete against each other seems a little contrary to the principles of the show. It seems at odds with a world that talks about how there is no unemployment and everyone is able to do what they set out to do. If all the applicants were capable of being a Starfleet officer, I would think that they would accept them.
As per usual, I am probably over analyzing it. This aspect of the plot is probably exists just to facilitate the moral (see Spoiler Section below). If any of the discussion above sounds interesting, go ahead and watch it. Otherwise, it is pretty safe to skip.
By the end of the exam, it is clear that Wesley and Mordock, another candidate, are neck-and-neck. Wesley takes a moment during one of the tests to help his opponent, which causes him to fail to get accepted into Starfleet. Picard delivers what I assume is the moral of the episode in this statement: "You have to measure your successes and failures within." Written down it's a little bit trite, but it is no worse than the life lessons that other shows of the time dole out (I'm looking at you Full House...).
(1) A few episodes back, I talked about a gesture that Patrick Stewart often makes which is jokingly called "The Picard Maneuver". In this episode, we almost see "The Riker Maneuver" (I think it is called that). As you continue to watch the show, you may notice that Riker often straddles a chair. This action denotes to us, the audience, his laid-back persona. He doesn't quite do it here, but he does swing his leg over the chair like he is going to, but without turning the chair around.
(2) This episode also features the first instance of Riker ALMOST getting to command the Enterprise. Picard tells Riker that he has been offered an admiralty, and you can see the excitement in Riker's face until Picard tells him that he has turned it down. Throughout the show, there are a number of times where Riker seems to waiting for the chance to command the Enterprise. Sometimes he gets to command of the ship for a little while. At other times, he is offered a position somewhere else and turns it down. He just can't seem to catch a break.