Even if their relationship is strained, Vito won't take a family photo without Michael. His unconditional love for his family is reinforced later when Vito is touched upon hearing that his godson Johnny Fontane has come all the way from California, it doesn't matter if he's only showing up to ask a favor.
Clemenza telling Michael that they were all proud of everything he did during the war, including his father.
After Sonny's death at the tollbooth, Don Vito has Tom call the mortician Bonasera to repay the favor he owes. Bringing Sonny's body to funeral parlor, Vito begs for the favor (doubles as a Tear Jerker):
Vito: I want you to use all your powers, and all your skills. I don't want his mother to see him this way. (he pulls away the sheet to reveal Sonny's bullet-riddled corpse. After a moment, Vito begins crying)Look how they massacred my boy...
Subverted in Part II, near the end when Mama Corleone passes away and Connie talks Michael into forgiving Fredo for his accidental betrayal. Michael comes out of hiding and hugs his grieving brother... only to coldly look up toward Al Neri, who nods quietly back to his Don. This is the Darker and Edgier sequel: Michael's not about to forgive anyone...
A sweet moment during this scene is when Vito initially puts the piece of orange in his mouth. Little Anthony seems to get scared and starts to cry and Vito quickly reassures him with a comforting hug.
In the book, when Vito is stricken, he urgently sends Anthony away. Not to get help, as Vito knows it's pointless, but so that Anthony won't see his grandfather actually die in front of him.
Michael and Sonny's last scene together before he goes off to kill Sollozzo and Sonny's promise to let Kay know when the time is right.
Vito saying to Michael "I never wanted this for you." combined with their prior discussion of Michael's 3-year old son reading the funny papers.
After Sonny's death, Hagen is shaken and goes to pour himself a drink. Don Vito walks in and Tom scrambles to the effect that he was about to waken him and let him know. Vito initially reproaches him, but later on, he embraces Tom.
"You've been a good son. You comfort me." Telling Hagen that he had acted properly in this terrible time.
The hospital scene in Part I. Vito is vulnerable and set up to be assassinated, and the only person there to help him is Michael. When he sees how much Michael cares for him and wants to protect him, he silently begins to tear up.
In Part II Grocer Abbandando is forced to give Vito's stockboy job to Don Fanucci's nephew. Abbandando is distraught when giving him the news but Vito tells him he understands and says he'll never forget how he took him in when he first arrived in New York when he was a boy. On the street Abbandando tries to give Vito a box of groceries for him and his family Vito politely refuses. When Vito comes home he places a pear he bought on the table, which is enough to make his wife happy. As they sit down to eat dinner Vito kisses her.
In the movie commentary, during the baptism scene, Francis remarks how blessed he was to have Sofia born to him at the time and pauses to note how cute she was.
Sonny's relationship with Tom in general. They have a heated argument over whether or not the family should go after Tattaglia. Sonny says something offensive to Tom claiming that, he wouldn't be so riled up if he had a War time Consigliere that was Sicilian. When he realized what he said, Sonny apologizes and offers Tom a seat at the dinner table even putting an arm on his shoulder reassuringly. Sonny is, after all, the whole reason Tom is a Corleone.
There's a moment in the book after Tom is released from his kidnappers. Sonny rushes to grab him in a hug, thus revealing how much he loves Tom and how genuinely afraid he was for his safety. Observing this, Michael realizes that Sonny is closer to Tom than he'll ever be to him, his biological brother.
Michael's discussion with Tom after the bedroom shooting. Michael reassures Tom (who since halfway through Part I has thought Michael's kept him and his talents at arm's distance) that he's kept things from him to protect him because Tom is his most trusted associate and his brother. Mike then makes him Acting Don while Mike sniffs out the traitor. Tom is so overwhelmed by emotion over realizing how much Mike genuinely loves and respects him that he's almost in tears.
Sonny's intense protectiveness over his baby sister in regards to her abusive husband. When Sonny sees her bruised face for the first time, his expression is pure "Must. Defend. Sister." It's sharply contrasted with Vito, who feels the honorable thing to do is to leave the marriage alone. Sonny doesn't care about honor, he cares about family.
In the first film, when Michael is outside the hospital trying to protect his father from being assassinated, Enzo the baker arrives to visit Don Vito. When he hears that Vito is in danger, he refuses to leave, even though he is clearly scared, and stands outside the hospital with Michael to bluff the would-be assassins. Vito may be a criminal, but it's clear plenty of people in the community love and respect him.
Michael: You better get out of here, Enzo, there's gonna be trouble.
Enzo: If there is trouble, I stay here to help you. For your father. For your father.
From part II, one that's easy to miss and might depend on your interpretation; Neri's attitude towards Michael ordering Fredo's death might be just cold professionalism, or it could be a matter of that he'll do it but won't enjoy it, and might not even like the order in the first place. Take note of the fact that when Fredo and Neri are out on the lake, Neri could take out Fredo at any time, but opts to wait until Fredo finishes his Hail Mary.