In the third part of "The Ultimate Doom", Sparkplug thanks Spike and the Autobots for coming back to Cybertron to save him from the Decepticons after getting brainwashed, to which Spike replies "It's like you taught me, Dad: You never quit on the people you love."
Brawn lampshades this by retorting "Mushy, but true..."
When Grimlock reconciles with Optimus Prime at the end of "War of the Dinobots" and shakes his hand in friendship for the very first time:
"Me Grimlock was jealous of Optimus Prime."
When Skyfire is found frozen in the Arctic in "Fire in the Sky" by the Decepticons, they take him back and thaw him out at their base. Starscream takes over the efforts to reactivate him and acts very concerned, explaining that he was a friend of his before the war began. Now Skyfire, being naive about the war and rather big and powerful, would be a valuable ally against Megatron, and it could be argued that this was Starscream's motivation. However, not only did he show no signs of an ulterior motive, he sounded genuinely concerned, and didn't bother trying to mask it with arrogance or bravado. Starscream rarely if ever demonstrated real concern for anyone but himself before or after this episode, so this moment in particular was very touching
In the episode "Heavy Metal War" Megatron challenges Optimus Prime to a battle where the loser and their faction will be exiled from Earth forever. Since no one is allowed to interfere with their duel the rest of the Autobots and Decepticons instead get themselves good seats for the fight. This leads to a blink and you'll miss it moment where Soundwave ejects both Ravage and Laserbeak and then proceeds to petRavage.
Hoist and Grapple in "The Master Builders" manage to show a deep friendship in the span of only thirty minutes, especially the last lines of the episode.
Hoist: There'll be other towers to design and build, Grapple.
Grapple: It was magnificent, for a little while. Wasn't it, Hoist?
Any time Bumblebee gets a little recognition. He's always so grateful, but in a humble way.
In "Dinobot Island", Bumblebee wraps up a call with Optimus Prime and gleefully remarks to himself, "I didn't know (Optimus) thought so much of me!"
It's not really touched upon in the cartoon, but in almost all G1 continuities, Bumblebee feels insecure about his size, and is always trying to impress the Autobots he looks up to. What he doesn't realize is that because of his endless determination they look up to him.
A misplaced one, but Peter Cullen has often said one of his favorite lines from the show was "Hold onto your dreams... they're the stuff the future is made of..."
What he doesn't remember is that line was from "The Core", one of the poorly animated AKOM episodes, and that that "dream" was keeping Devastator under Autobot control. Beautiful line out of context, though...
Even within context, it regarded Chip being upset he couldn't convince a fully conscious Devastator to willingly join the Autobots afterwards, thus the dream is at least some hope the heroes can redeem the Decepticons. Perhaps as a two way case, Devastator only refused point blank because he didn't wish to betray "noble Megatron".
Megatron: I do this only for my fellow Decepticons. It grieves me that you may also profit.
In the same episode, before forming Devastator to stop the drill that threatens to destroy Earth's core,Scavenger shares a moment with fellow constructicons, telling them, that he was glad to work with them.
How, in "Make Tracks", Raul goes from being a small-time car thief and hoodlum to a friend of the Autobots, namely Tracks and Blaster. He is one of the few non-Witwicky related humans to reocccur in the series.
The Aerialbots creating a Stable Time Loop by saving Orion Pax in the past in "War Dawn". Keeping in mind it wasn't until after his resuscitation they found out they had saved the future Optimus Prime, one act of nobility to whom they thought was a complete stranger strengthened their future. Megatron wounding him in the first place was enough to break all of their admiration over the Decepticons' skills.
Slingshot, in particular, goes from being in awe of Megatron and the Decepticons at the beginning of the episode to despising them after witnessing them attack the Energon Storage Facility and mortally wounding Orion Pax.
Slingshot: "You'll NEVER get away from us, Megatron! We won't stop fighting until you're finished! You hear?!"
There's a brief scene in "Ghost in the Machine" where Bumblebee and Spike team up again. It's the only time it happens in the third season and is a nice nugget for longtime fans who have seen them be friends from the beginning. 'Bee even says, "Just like old times."
Galvatron: Uhh... Cyclonus... they'll destroy me... help me... uhh... HELP ME!!!
It takes about two seconds before Cyclonus is convinced it's time to act.
Cyclonus: No, release him! Let him go!
To put it into context with his loyalty, throughout the episode his main concern was Galvatron and openly questioned the Doctor's orders and while going through the ream of paperwork he looked back at the cage carrying his unconscious leader away. He was willing to risk his life if it meant Galvatron could be cured of his insanity and after the Ayla Solution backfired (as mentioned before he tried to save Galvatron from the lobotomy) he stayed by Galvatron's side and admitted it was his fault (he got hit for his troubles) and followed Galvatron into the crevisse where he destroyed the Memory Core. That is loyalty.
Cyclonus was the only Decepticon not to leave Galvatron's side in 'The Burden Hardest to Bear'.
Much like "War Dawn", the Autobots working with a rookie, A3, thrust into the future, only afterwards finding out they had helped the future Alpha Trion.
A lot of the interaction between Daniel and Arcee, particularly in "The Rebirth". Her motherly nature towards the boy can be the source of much W.A.F.F..
The last 30 seconds of "The Return of Optimus Prime, Part 1" is the Heartwarming Moment of Transformers.
Sky Lynx: Our leader is back.
Optimus: Yes, Sky Lynx. And this time, no force in the universe can stop me.
While it's mostly due to the limited 80's writing, it was great to see that Hot Rod held no grudge against Optimus Prime being made leader of the Autobots again after he gets the Matrix back in "The Return of Optimus Prime". Hot Rod doesn't care about losing his role as leader, he's just glad to have Optimus back.
Especially poignant when you remember that Hot Rod was partially responsible for Optimus' mortal injury during the Movie. He carried that knowledge with him all this time; now Optimus is finally back and his guilt is a little lighter.
In a minor case, Optimus Prime seemed rather unwilling to take the Matrix from Rodimus in the first place, suggesting he genuinely wanted to respect his current replacement.
Even Galvatron was moved by Optimus managing to save the universe:
In "Fight or Flee", Sandstorm asks if something is wrong with Blurr after finding out that he talks faster than normal. Kup responds by putting a hand on Blurr's shoulder and saying that that's just the way he is.
In this commercial, where Optimus seems very happy to see his old teammates. (Unfortunately, Starscream has to spoil it.)
After Daniel is wounded in a Decepticon attack, Arcee volunteers to take part in the Headmaster process in order to restore his ability to walk.
Daniel and Arcee bonding as a mother and child would once the Headmaster process is complete.
Come to think of it, where was Carly during all this, anyway?