Wesley Crusher takes the Starfleet entrance exam, pitted against three other young candidates who are as brilliant as he is. Only the highest of the four scores will go to Starfleet Academy. This is the sort of story that, at age 12, made me fear for my future of entering high school and college. Consider — here were four fictional characters who were far more brilliant than I was, and three of them would be going home as failures, despite their brilliance. Now there's a frightening message about competition for a 12-year-old. Guess you'd better study harder, kids.
Finally, this is a Wesley-oriented storyline I can tolerate. The reason it works is because it treats Wesley as a teenager instead of the crazy kid who saves the ship with his implausible genius. It treats him as a young person who has a lot to learn about life. Yes, he may be a hopeless geek (and still annoying), but at least the story recognizes him for his human qualities rather than his techo-plot ones (the "stress test" at the end deals with his own personal issues rather than his warp theories).
Meanwhile, on the Enterprise, Picard's old friend Admiral Quinn (Ward Costello) sends in his investigative pit bull, Lt. Cmdr. Remmick (Robert Schenkkan), to look for problems in Picard's command. Remmick interrogates the entire bridge crew, pissing off everybody in the process. This leads to some pretty good scenes of conflict on a show sometimes notorious for its lack of interpersonal conflict. The investigation is dramatically on shaky ground because the episode never says what Quinn and Remmick are looking for (except "problems"). In the end, Quinn levels with Picard about a possible conspiracy within Starfleet, and offers him a promotion. It's a strange, albeit watchable, series of notions, interviews, questions, and conclusions.
Although it doesn't have a strong driving focus, "Coming of Age" is about the personnel workings of the Enterprise crew more than it's about a generic plot, which is in its favor.