The scenes where Dave visits his former wife and (especially) his dying mother.
Floyd's letter to his son when returning from his mission:
Your children will be born in a world of two suns. They will never know a sky without them. You can tell them that you remember when there was a pitch black sky with no bright star, and people feared the night. You can tell them when we were alone, when we couldn't point to the light and say to ourselves - 'There is life out there.' Someday, the children of the new sun will meet the children of the old. I think they will be our friends. You can tell your children of the day when everyone looked up and realized that we were only tenants of this world. We have been given a new lease and a warning from the landlord.
Also him bonding with his son before the mission, and the last scene with his wife.
I'll be scared enough for the both of us.
The scene where Zenia, in a panic before they attempt aerobraking, comes to Floyd's cabin. The two cling to each other as the Alexi Leonov shoots like a fireball across Jupiter.
Also the peck on the cheek she gives him afterwards.
The bonding of the crew of the Alexi Leonov and their american passangers in general is very heartwarming.
This bit of dialogue between two "former foes":
Hal: I'm afraid. Dave: Don't be. We'll be together. Hal: Where will we be? Dave:Where I am now.
ALL THESE WORLDS ARE YOURS EXCEPT EUROPA. ATTEMPT NO LANDING THERE. USE THEM TOGETHER. USE THEM IN PEACE.