FINN: Well, this is what we look like. Some of us. Others look different.
REY: BB-8 says he's on a secret mission; he has to get back to your base.
FINN: Apparently he's carrying a map that leads to Luke Skywalker, and everyone's after it.
This is the call to adventure as clearly as it can call. That's an easy sign of a character coming of age: being tasked to complete something super important.
REY: I didn't know there was this much green in the whole galaxy...
Another good sign of coming of age: seeing somewhere new and going somewhere you've never been before. It broadens the horizons in more ways than one.
HAN: You might need this.
REY: I think I can handle myself.
HAN: I know you do. That's why I'm giving it to you. Take it.
Notice that coming of age isn't portrayed as an all-at-once thing in this film. It's a series of little steps that don't seem like much at the time, but which add up to something huge.
REY: Are you offering me a job?
HAN: I wouldn't be nice to you. It doesn't pay much.
REY: You're offering me a job.
HAN: I'm thinking about it.
Hey, what's more coming-of-age than a job? And a job on the Millennium Falcon? Who could pass that up?
MAZ: That lightsaber was Luke's. And his father's before him. And now, it calls to you!
This means more than just accepting a job or seeing through a task. The Force is attempting to communicate with Rey not because she needs its guidance, but because it needs something from her.
KYLO REN: She's just beginning to test her powers. The longer it takes to find her, the more dangerous she becomes.
The clock is ticking, it seems, which suggests that coming of age is a finite process. Kylo Ren wants to stop Rey not only to corrupt her to the dark side, but possibly to save his own skin…since she may be stronger than he is.
FINN: He took her! Did you see that? He took her. She's gone!
HAN: Yeah, yeah, I know...
This is a little odd: it may actually be that Han is coming of age. He's in his 70s, and he's still pretty much running away from his mistakes. By acknowledging that his son has taken Rey away, he's acknowledging that he can't just ignore the rift between them anymore. He has to pay the piper for being a not-great father.
KYLO REN: We're not done yet.
REY: You're a monster!
KYLO REN: It's just us now. Han Solo can't save you.
It all comes back to facing your demons—the guy in the scary mask who makes you wet your pants with fear. Look that evil dude right in the eye and get ready for a fight. We all have to face him sometime, and when we do, we're just a little bit stronger for it.
KYLO REN: That lightsaber. It belongs to me!
FINN: Come get it.
The last step to coming of age is standing on your own, alone. Finn manages to do quite well with the moment, even though the lightsaber isn't his, either.