The Turtles' reunion with Splinter at the end of the first movie
Just before the fade to the credits, while Leo, Donny and Mikey are dancing in celebration, Raph, who throughout the film was the most visibly shaken by Splinter's abduction, can be seen embracing his master again.
Not to mention Leo keeping constant vigil with Raphael the whole time he's unconscious and the "Kodak Moment" when Raph finally wakes up.
Ralph returns it when he cuts off Leo's apology in a heartbeat. "Leo...dont."
Splinter: I am proud of you, my sons. Tonight you have learned the final and greatest truth of the Ninja: that ultimate mastering comes not from the body, but from the mind. Together, there is nothing your four minds cannot accomplish. Help each other, draw upon one another, and always remember the power that binds you. The same as what brought me here tonight, that which I gladly return with my final words: I love you all, my sons.
Splinter later helps Danny redeem himself by telling him that "...all fathers care for their sons."
The typical lecture that Raphael expects from Splinter instead becomes a very moving scene in the first movie. The somber music and Raph's speechlessness help to double this moment as a bit of a Tear Jerker.
Splinter: [Anger] is an unconquerable enemy. You are unique among your brothers, for you choose to face this enemy alone. But as you face it, do not forget them. And do not forget me. I am here, my son.
At the end of III, Mikey's a bit depressed because he had to leave the friends he made in feudal Japan, coming to the conclusion that Growing Up Sucks. How does Master Splinter help his son out of his funk? By wearing that lampshade from the beginning!
Raph hands Leo a pair of katanas to replace the set which were broken in their fight Raph: You're gonna need these if you're gonna lead us out of here. Leo: I'm gonna need you, too.
In the same movie, Leo coming back from his pilgrimage and being welcomed by Splinter. Leo and Splinter calling each other Father and Son, respectively, can make any fan of the series shed happy tears.
Turtles Forever when all three turtle groups congratulate each other at the end of the film.
Leo: Bros? Hey! We're bros!
Casey Jones in the first movie. First off he barely knows Raphael save for one fight with him, however after spotting him in danger he jumps into action. He didn't even meet the other four but stalled the foot long enough for them to get to a safe distance. And that is before he was established as their friend.