It's so sad you can even cry after your first playthrogh.
The fact that this was basically the first time we see ourselves talk to our partner. Hearing the good-bye coming straight from the player...
You: *starts to glow brighter* '...! The light...The light is getting brighter...'
Partner: "D-Don't! Don't...Don't go..."
You: "Thank you for everything, [Partner]. I'm glad we got to train together at the guild...I'm glad we got to go on adventures together...I'm glad...I got to know you [Partner]. "
You: "I'm sorry. I'm so lucky that you were my friend..."
Partner: "I feel the same way! To me, you're...more important than anything..."
You: "Yes...I feel the same way. [Partner]...Even after I disappear from here...I will never forget you... *glows brighter and disappears...*"
Partner: *calls your name many times and collapses on the ground in tears*
Dialga, his sanity restored, feels the lingering sorrow of one of the heroes who saved the world, and... "I GIVE YOU... THE FUTURE! THIS IS MY THANKS! PLEASE ACCEPT IT!"
The second ending (for the post-credits story) as well as when you reunite with Manaphy.
Finding out Grovyle, Celebi and even Dusknoir survived in Sky. The music really cinches it.
And the scene right before that is probably one the best in the game. Grovyle holds a crying Celebi in his arms as the two watch the sun rise while they - and the world around them - fade into oblivion.
Not to mention the fact that not only Dusknoir, but also his Sableye mooks, have all pulled a Heel Face Turn by the end.
A somewhat obscure example concerns Team Tasty, consisting of an oblivious Swellow and his Wurmple partner who is highly distressed, thinking his leader wants to eat him. Around the time of the Manaphy chapter, Wurmple is alone and wondering what to do, before deciding he should put his trust in Swellow and confront him about his concerns. They aren't seen again until right after Aegis Cave has been unlocked/explored for the first time, and they're not even in town, instead on the beach. After their conversation, there are no longer any fears about where they stand.
Swellow: What...what are you suggesting? Me? Eat you? Are you kidding? Wurmple! You're my trusted partner. You're my one...and only partner. There's no replacing you! You and me, we're going to keep on exploring together like we've always done. (Pause - Wurmple tears up) Hm? Why are you crying?
"Truly bad Pokemon...don't really exist anywhere"
Even Darkrai isn't immune, as after he loses his memories he becomes nice.
When the guild master says that very nice speech before the Darkrai battle.
"Bidoof's Wish". He asks Jirachi for a friend he could be a sempai/mentor to. Some time later, enter the player characters. Awww...
Not to mention when he's about to be mugged by Team Rogue, and the entire guild shows up to defend him. There's even a second (obviously, much easier) fight scene, with a remix of the Guild music playing through it.
From "Igglybuff The Prodigy". Igglybuff is screaming and crying over the arrest of his master, Armaldo, and is about to attack Magnezone and Nidoking. Armaldo himself stops him and calms him down, accepting his own arrest and giving Igglybuff the very first treasure they found together to remember him by. Also a Tear Jerker.
Chatot gets one of his own later during his Heroic Sacrifice. That, and the entire Brine Cave cutscene with him and Wigglytuff. Awww.
When you and your partner return from the future and decide to get help from the guild, all the guild members emerge and sort of gape at you in shock. Then Wigglytuff himself shows up, and what's the first thing he says?
Not to mention the chapter immediately following that scene, The Guild's Crew. Chatot laughs at your story, but then Bidoof steps up and announces that he believes you, causing the rest of the guild to do the same. And then Wigglytuff and Chatot reveal it was a Secret Test of Character for the guild, and finally, the entire guild unanimously decide to assist you at all costs. That entire segment is heartwarming on So. Many. Levels.
[Partner's name] told this story as often as possible, to as many Pokemon as possible. It was a story that needed to be told. It was a story of hope for world peace. It was a story of hope for future peace.
Turning to a new recruit and having them tell you they feel safer with you.
Finding a Sky Gift and being able to give it to any of your guild members, including your partner. And when you yourself receive a Sky Gift from someone, the game says you get 'a warm fuzzy feeling' inside.
How The Player breaks free of Darkrai's illusion. By realizing that their partner would never give up like they did in Darkrai's illusion.
The moment when you, your partner and Grovyle all return from the future, in Sharpedo Bluff, when Grovyle expresses how happy he is to see you again. Sure, many people say you and Grovyle only teamed up just to save the future, but this moment really showed how much he truly cared for you.