The entire plot is kicked of by Marlin not wanting the last thing his son says to him to be "I hate you." What's one of the first things Nemo says when they reunite?:
Nemo: Dad... I don't hate you.
Another one: "I love you, Dad." "I love you too, son."
This is also a great scene because Nemo actually swims back to give his dad a hug when, earlier in the film, he'd been feeling a little stifled from his father being overprotective.
Dory:when I look at you...I can feel it. And - and I look at you, and I...and I'm home! Please... I don't want that to go away. I don't want to forget!
After they have escaped from a hungry shark and massive minefield explosions, Marlin and Dory are exhausted. Marlin is anxious to find his missing son, Nemo, but now he has lost his best clue for finding him — a scuba mask inscribed with the address of the diver who captured Nemo. Dory helps Marlin find hope. Discouraged, Marlin says, "That was my only chance at finding my son; now it's gone!" But Dory is not so easily deterred. "Hey, Mr. Grumpy Gills," she says. "When life gets you down, you know what you gotta do? Just keep swimming, just keep swimming, just keep swimming, swimming, swimming. What do we do? We swim, swim."
The fish fighting against the net. "Keep swimming! Keep swimming!" Especially when you realize it's Dory's little "Just keep swimming!" motto. The idea that united forces and sheer will can overcome the dangers of the world and win out over fear.
The scene where Nigel tell Nemo that his father is looking for him and recalls everything he's gone through to do so. That music...it gets you every time.
Even more awesome because Nemo originally thought his father a coward. The fact that if his Dad was willing to risk life and fin to find him, then dammit, he was gonna jam that filter!
A crowning moment of awesome is how Nigel even learns of Marlin's exploits. Marlin's heroics and absolute determination to find his son as gained fame and spread across the ocean. From swordfish to seagulls, there's an entire ocean rooting for the two clownfish to be reunited.
When Marlin and Nemo finally reunite. Especially when Marlin realizes he's still alive after mistakenly thinking that he was dead.
And before that, when Dory remembers who Nemo is and just tackles him, spinning him around ecstaticly. She's just so happy to see he's okay!
You know, to give the guy his due, Dr Sherman didn't maliciously take Nemo. He genuinely believed he was struggling, even dying from being separated from the reef, and so decided to save him. Sure, he was wrong, but he had no way of knowing that.
How everyone in the Tank Gang (well, except for Gil, but he defrosts soon enough) immediately goes out of their way to make sure Nemo's alright and make him feel welcome, realizing he'll probably be scared and upset upon arriving in the tank.
Marlin: No, I promised him I'd never let anything ever happen to him.
Dory: Huh, that's a funny thing to promise.
Dory: Well, you can't never let anything happen to him, then nothing would ever happen to him. Not much fun for little Harpo.
There is actually no outright villian in this movie. Their main obstacles are situational and creatures that are out to get them are acting on instinct. Even the humans weren't intentionally trying to hurt the fish.
Meta and out of canon
Everyone wanted a Finding Nemo sequel, so Pixar decided to make it.
One that also doubles as a serious Tear Jerker: Several years ago, Ellen invited a young girl named Leah, who was undergoing cancer treatment, onto her talk show. When Leah was asked how she keeps positive about her treatment, she replied, "A little fish told me to just keep swimming!" After Leah died in 2013, someone on Twitter suggested that they dedicate the aforementioned Finding Dory to her, because "she really knew how to just keep swimming."
Marlin and Coral interaction on the deleted scenes from the Blu-Ray release where we see how they meet and their life as a married couple, also serves as an Tear Jerker because you know what happens eventually.