In the first episode, it's clear that Bruce is too old to be Batman anymore, and when he has physical problems chasing after some petty thieves, one of them stops and asks Bats with genuine concern, "Hey, are you okay?"
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. 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.
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 Mel 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...
In the episode where 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 on some level he's forgiven Freeze.
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 the episode "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 Maxine 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 dad.
The end of "Mind Games", when 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.
Clark Kent & Bruce meeting for the first time in years, complete with Terry watching on.
Bruce: What're you smiling at?
In one episode, a rogue AI is making all computerized things in Gotham- which is almost everything- go berserk. Terry discovers this while on a date with his girlfriend and they end up trapped on an elevator. After he stops said 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 Terry and Dana escape from Packrat's lair(under the guise of Batman), Terry shows up and starts to apologize to Dana (whom was mad a 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 them both 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.
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.
In Zeta's second appearance, he's come back with a close human friend who knows who he is and deeply cares for him.
In "The Last Resort", oddly enough, 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 "Disapearing 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 tells us 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.
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.