In the first episode, it's clear that Bruce is too old to be Batman anymore, and when he suddenly has a heart attack while battling some petty thieves, one of them stops and asks Bats with genuine concern, "Hey, are you okay?"
There's an absolutely adorable scene in Heroes where Matt and Terry are at the mall and Matt is looking at Terrific Trio merchandise when a store clerk indirectly insults Batman by saying it's "nice to have some good, wholesome heroes the kids can look up to", causing Matt to say that the Trio's nothing compared to Batman (not realizing his big brother is in fact Batman). This prompts Terry to buy Matt some ice cream.
Terry convincing Bruce in the pilot to let him stop Powers from shipping the nerve gas, and to confront his father's killer.
Terry: I read up on you, Mr. Wayne. I know how you lost your folks. The guy who murdered my dad is on that hover transport. This is my one chance to nail him. (beat) Bruce: ...the hoverpad is in the northeast sector. Terry: Thanks. Wish me luck. Bruce: ...good luck.
A small one in the episode "Splicers" where Terry and Bruce's dog Ace have been having trouble getting along, but then Ace tries to save Terry while they are fighting a mutated Dr. Cuvier: "DON'T. TOUCH. MY. DOG."
Another Ace moment: After being attacked by the resurrected Joker, Ace recovers by watching classic WB cartoons.
"Babel": Bruce's willingness to give himself up to Shriek in Terry's place is definitely an "AWWWWW" moment. Also, this:
Bruce: That kid's done a lot for this city. It's time for this city to do something for him.
Bruce's final goodbye to Talia in "Out of the Past." Might also count as Tear Jerker depending on the viewer.
Also from the beginning of "Out of the Past": Bruce only storms out of the musical making light of his career as Batman when the play begins making a mockery of Jim Gordon.
Fitting, considering it was the end of the series (until "Epilogue"), was Bruce's speech to Terry at the end of Return of the Joker:
Bruce: Terry...I've been thinking about something you once told me. And you were wrong. It's not Batman that makes you worthwhile; it's the other way around. Never tell yourself anything different.
The ending of the episode "King's Ransom" when Melanie bails out Jack and gets him a job.
Too bad Jack screws it up in the comics by reforming the Royal Flush Gang with himself as King. That may not be canon, though...
Don't worry, it's definitely non-canon, given the weird continuity of the comics.
In "Meltdown," where Mr. Freeze returns, Bruce is convinced that Fries is still a bad guy. Throughout it, he's warning Terry that Freeze is no good, but Terry stands up for him. At the end of the episode, after all he's seen, Terry says that Bruce was right... but Bruce says they were both right, indicating that he now believes that Freeze sincerely intended to reform and that on some level he's forgiven him.
Earlier in the episode, Freeze gets attacked by a assassin whose family had been killed by Freeze decades ago before Terry stops him. Terry is about to take him to the police before Freeze tells him to let him go, saying "I've done this man enough harm."
Freeze's final words "Believe me...you're the only one who cares...", when you remember from his first appearance in "Heart of Ice" to this point, both Terry and Bruce had been the only ones who genuinely cared about Freeze's well-being.
Then to top it off, Freeze last act is to save a life. Creating a barrier of ice to prevent Terry from trying to help and forcing him to flee.
A small one that is in the episode "Curse of the Kobra: Part II," Max has been kidnapped by the Kobra cult to be their queen and Terry is desperate to find her despite his loads of injuries, including four cracked ribs that hurt when he breathes. When he and Bruce finally locate her whereabouts, there's this exchange:
Bruce: What about your breathing problem? Terry: (pulls on the mask) I'll breathe on the weekend.
Terry and Melanie's Caught in the Rain moment. Usually moments like this are stupid...but the music and their expressions, they were very genuinely happy to see each other. It was just too sweet!
The first time Bruce called Terry by his first name. It hits you right then just how much Bruce cares about Terry and keeping him safe.
The bro moment between Bruce and Terry at the end of "Dead Man's Hand," after Terry watches Melanie/Ten get hauled away to prison:
Terry: This kind of thing ever happen to you?
Bruce: (smirks) Let me tell you about a woman named Selina Kyle...
In "Final Cut," Terry is threatened with a bomb that could level an area of Gotham if he doesn't keep a member of the Society of Assassins alive. When Max is attacked by Curare, Terry, for the sake of a friend, risks all of Gotham to save her, knowing that, in his absence, Curare could easily get to the man.
The end of the episode after they both save the day:
Max: How do you do it?
Terry: (dabbing a wound on her forehead) It's easier for me. I've got the suit.
Max: No, how do you save the world and not tell anybody?!
Terry: I can tell you. You can tell me. That's why I'm glad you found out, partner. (they grin at each other and the episode ends)
The end of "Revenant" where Terry and Matt are going through pictures of a trip they took with their late dad.
The end of "Mind Games," when Tamara, the little girl Terry saved, sends a psychic projection to him, just waving and smiling.
There's also a moment midway through the episode where Terry goes into Matt's room and pulls up his blanket over him as he sleeps. Aw.
Bruce and Superman meeting for the first time in years, complete with Terry watching on.
Bruce: What're you smiling at?
In "Lost Soul," a rogue AI is making all computerized things in Gotham (which is almost everything) go berserk. Terry discovers this while on a date with Dana and they end up trapped on an elevator. After he stops the elevator, he is frantically fiddling with the controls in order to get down to the ground quickly so he can go check on Bruce.
At the end of "Rats," after Batman and Dana escape from Packrat's lair, Terry shows up and starts to apologize to Dana (whom was mad at him for missing a date earlier) until Dana hugs him and tells him to shut up.
The entire "The Trigger Man" arc is about Mad Stan trying to reclaim his lost doggy. Preferably, with large quantities of explosives, but still. He gets caught in the end, but is allowed two visits a month with his dog, which is implied to be Batman's work.
The end of "Hooked Up" when Max apologizes for getting both herself and Terry/Batman in trouble, and Terry forgives her.
Max: I just wanted to be a part of something.
Terry: Well, congratulations, because you're a part of it now, but we'll take it a little slower next time.
In "Ace in the Hole," after Terry finally brings Ace back from a dog-fighting ring, Bruce is so clearly relieved even though he'd spent the episode telling Terry not to worry.
From the same episode, the flashback of Bruce bringing Ace home in the first place. It's especially heartwarming given that it happened during the reclusive years between Bruce hanging up the mask and passing the mantle to Terry; as bitter and alone as he became, he remained basically good-hearted.
In Zeta's second appearance "Countdown," he's come back with a close human friend named Rosalie who knows who he is and deeply cares for him, as seen in The Zeta Project.
Mad Stan sees a human-looking Zeta being attacked by federal agents, and immediately jumps in to rescue him. The guy might be a lunatic and a terrorist, but he can care about people.
In "The Last Resort," when Terry talks with Chelsea while she's in the correctional facility. Despite him being a pretty bad boyfriend for her best friend, she's still glad to see him because of all she's been through, and he still cares about her (despite them barely interacting).
While virtually all the episode with Eggbaby is played for funny all the moments, the ending where it turns out Terry gets the best grade out of everybody cause he was the only parent who provided enough "intellectual stimulation" for his eggbaby in addition to meeting its basic needs. It doesn't hurt that Dana gave him a very heartfelt little smile to boot, either.
In "Disappearing Inque," Bruce comes to save Terry when Inque captures him and threatens to kill him if Bruce doesn't show up. He does it in the exosuit that puts dangerous levels of stress on his heart, as he explains earlier in the episode. Terry's clear dread and panic when he sees the suit in action, and when Bruce finally goes down, make it even more heartwarming.
Plus, considering events revealed in ROTJ, this becomes even more powerful when you realize Bruce is trying to prevent a second tragedy...
In "Out of the Past," Bruce reminiscences his glory days on his birthday...watching photos of the past women of his life on the Batcomputer, including Zatanna, Lois Lane, Selina, and Barbara. Unfortunately, due to this being made before Justice League, Diana isn't seen at all.
Terry, at one point, mentions the "Near-Apocalypse of '09," and that Talia turned on her father and aided Batman in putting an end to Ra's scheme. If what he's referring to is Ra's and Talia's last appearance in the Superman episode "The Demon Reborn" (in which Talia showed "no" sign of reforming), it could be that Bruce made arrangements after her apparent death that she be remembered in history kindly.
Terry: So how mad was Dana that I blew off our date?
Max: Actually, she wasn't mad at all. Since she understands the "special nature" of your job.
Terry: What?! She found out?!
Max: Relax. She was talking about your relationship with Mr. Wayne. She says that with your dad being gone, it was important for you to have a positive father figure. Bruce Wayne. Some father figure.
Terry: (smiles) Maybe she's not so far off.
Near the end of "The Call, Part 2," Starro and his kind are allowed to go home because, as Superman himself puts it, they never asked for what happened to them. After what he'd been through, Superman's still the All-Loving Hero.
Starro himself is more than happy to cooperate; all he'd ever really wanted was to go home.
When the antagonist threatens to reveal Batman's identity on "Sneak Peek", Terry kicks himself for not checking the Batmobile for potential bugs. Bruce's response? It's far gentler than one might expect.
Bruce: I always assumed I'd be found out sooner or later. It's just galling that a piece of garbage like Peek had to be the one.
In the conclusion of the Justice Lords arc, T thanks Terry for his actions by arranging a surprise meeting with Warren McGinnis, who did not perish in the Lord timeline. What follows is a conversation where Warren apologizes for the current state of their relationship, hoping to make things better (of course thinking he was T). However, it is a statement which Terry needed desperately to hear, considering how their last interaction went the night Warren was killed, and wouldn't be far from what main timeline Warren would have said to Terry if he had the chance.
When Diana reunites with Bruce. She's still not used to the fact that it's been years since she's seen a Bruce or a Batman, nevermind him being an old man now. Bruce smiles wistfully and tells her it's okay, before they embrace.
The Bat Family finally reconciles in the final issue. It's most poignant with Dick finally forgiving Bruce after Dick's near-death experience.