In the script, John said "Sorry" to the guard as they drive pass him.
The Terminator hot-wires a vehicle... after which John instantly pulls out the keys from the visor.
John: Are we learning yet?
Which sets up a Brick Joke later when they commandeer another vehicle and the Terminator does check the visor for the keys.
After that, John teaching the Terminator how to speak in a more humanlike fashion.
John: No, no, no, no, no. You gotta listen to the way people talk. You don't say "affirmative" or some shit like that. You say "no problemo." And if someone comes off to you with an attitude, you say "eat me." And if you wanna shine them on, it's "hasta la vista, baby."
Terminator: (flatly) Hasta la vista, baby.
John: Yeah, or "later, dickwad." And if someone gets upset, you say "chill out." Or you could do combinations.
Terminator: (also flatly) Chill out, dickwad.
John: That's great! See, you're getting it!
Terminator: No problemo.
Explaining to Dyson about Judgment Day.
Voiceover: It's not every day you find out you're responsible for three billion deaths. He took it pretty well.
Dyson: I feel like I'm gonna throw up.
All things considered, he really did take it pretty well. Far better than most would.
"He's my Uncle Bob..."
After John finds out the terminator does whatever he says he asks him to stand on one leg. The scene follows with him standing on one leg whilst talking to some guys, until John quietly says "Put your leg down" out of the corner of his mouth.
He says it like such a mom to the tall robotic assassin.
"I need a vacation."
As Sarah is breaking out of the hospital, the guards try to reason with her and insist she's not a killer. Sarah snaps back they as far as she's concerned they're all dead anyway and they know how strongly she believes it, "so don't fuck with me!" Slow learners, this bunch...
When they break into Cyberdyne:
John: We got company. Dyson: Police? Sarah: How many? John: Uh, all of 'em, I think.
To make it even funnier, it is all of them. When the security guard asked them to send everything they had, they listened to him.
This in itself is both a bit of Fridge Brilliance and a Call Back. The police interviewing Sarah earlier in the film remind us that the first Terminator 11 years ago murdered several police officers. Of course they are going to send everything they have.
And even that's still not enough (but at least none of the cops die this time).
John needs money from the payphone to make a call. He asks the Terminator for a quarter. The Terminator smashes the bottom of the payphone, breaking open the coin collection box. Then he hands John exactly one quarter, as requested.
When the Connors are planning to flee to Mexico, the Terminator sees a baby, then picks it up and looks at it like he has no idea what it is.
This also subtly foreshadows some of the events in T4: leave it to SkyNet to be such a Smug Snake that it would program its killing machine to go silently taunting one of its worst enemies before terminating her.
John tries to explain what a smile is to T-800 - he then looks at a passer by and scans his smile to imitate it himself - this is the result.◊
John teaches T-800 to do high-fives and puts his own palm for T to swat it... with, basically, a metal rod. The T-800 doesn't break his arm, but we can see John smarting from it and saying "Owww...!" even though (since we're watching from Sarah's perspective) we can't hear him.
Afterwards, the T-800 pulls the same "too slow" trick John just taught him.
The security guard before right before the T-1000 kills him in an Eye Scream moment. He just got a full house on his coffee cup poker game. "Must be my lucky day!"
When the T-1000 spots John, he shoves the kid (John's fellow delinquent) who was trying to distract him out of his way halfway through his spiel. Blissfully unaware that he just had a close brush with death, the kid is insulted. "Hey!"
It's even funnier when you consider that kid is never, ever mentioned again. He basically got shoved out of the franchise!
A meta example: while filming the bar scene, a female passerby walked in on the set with Arnold in only his boxers. When she asked what was going on, Arnie replied that it was "male strippers' night". This becomes Hilarious in Hindsight in Terminator 3: Rise of the Machines when the T-800 walks into a bar that held a male stripper show.
"Say, that's a nice bike." Apparently, even the T-1000 thinks some bikes are cooler than others.
The escape from Cyberdyne.
T-800: *tosses a cop his grenade launcher* Here, hold this. *proceeds to rip the gas mask off the otherwise occupied officer's face.*
The final scene, where the T-800's hand sinks "thumbs up" into the fire...it's hard not to giggle at that.