McCOY: Admiral? Wouldn't it be easier to just put an experienced crew back on the ship?
KIRK: Galloping about the cosmos is a game for the young, Doctor.
UHURA: Now what is that supposed to mean?
Kirk is saying that it's time to hang up the spurs here, that maybe he and the gang need to find a nice desk job instead of saving the universe and such. In his mind, we may as well just roll the credits now. That's the attitude that his journey has to change.
KIRK: It had the virtue of never having been tried. Oh, by the way...thank you for this.
SPOCK: I know of your fondness for antiques.
KIRK: "It was the best of times, it was the worst of times." Message, Spock?
SPOCK: None that I'm conscious of...except, of course, happy birthday.
Interesting quote. We wonder if it's going to have some kind of payoff later on in the picture.
McCOY: Damn it, Jim, what the hell's the matter with you? Other people have birthdays. Why are we treating yours like a funeral?
It's a fair question, implying that Kirk's strengths lay in his virility and youth, and now that he's old, he doesn't have anything practical to contribute.
McCOY: Jim, I'm your doctor and I'm your friend. Get back your command. Get it back before you turn into part of this collection. Before you really do grow old.
Bones implies that youth lies in taking action, and that as long as Kirk sits behind a desk, his mojo will be gone. He's not entirely wrong, but Kirk getting back his command means a chance to learn new things to help him adjust, instead of reliving the glories of younger, studlier days.
SULU: I'm delighted; any chance to go aboard the Enterprise.
KIRK: Well, I for one am glad to have you at the helm for three weeks. I don't think these kids can steer.
The implication here is that age brings skill and wisdom that those young whipper-snappers don't have.
SPOCK: If I may be so bold, it was a mistake for you to accept promotion. Commanding a starship is your first best destiny. Anything else is a waste of material.
This is another way of saying you're only as old as you feel, and that if Kirk thinks he's still got it in him to go cruising around the galaxy, who cares how old he is?
SULU: Sir, you did it.
KIRK: I did nothing... except get caught with my britches down. I must be getting senile. Mister Saavik, you go right on quoting regulations!
There's a little humility in Kirk after all: the old Kirk (which is, ironically, the young Kirk) would never have admitted to an error like that.
KIRK: I did what you wanted. I stayed away. Why didn't you tell him?
CAROL: How can you ask me that? Were we together? Were we going to be? You had your world and I had mine. And I wanted him in mine, not chasing through the universe with his father. Actually, he's a lot like you. In many ways.
This is one of the most important pieces of dialogue in the film: Kirk has to face the consequences of the decisions he's made. That, more than the possibility of growing old, is what's eating at Kirk: the notion that he missed some cool things. Like a kid. He's got a kid.
And the kid's been doing some really cool stuff, and he missed it, because he was out spooning with space ladeez.
KIRK: There's a man out there I haven't seen in fifteen years who's trying to kill me. You show me a son and he'd be happy to help him. My son…my life that could have been…and wasn't. And what am I feeling? Old. Worn out.
CAROL: Let me show you something that'll make you feel young as when the world was new.
And by admitting he feels old, Kirk takes the first real steps toward not feeling so old any more. It sounds like pop therapy, but it's also an important stop on the Hero's Journey: the part where you confront and acknowledge your deepest fear in order to defeat it.
McCOY: You okay, Jim? How do you feel?
KIRK: Young. I feel young.
See? One of pop culture's most beloved characters can die in heartbreaking fashion, and we can still have a happy ending!