Most of my favourite moments seem to be involving Tom.
One that I like very much is of course from Part I, and that I don't ever see mentioned is when they are awaiting the call about which place Michael is going to meet Sollozzo at. They are discussing the 'meeting' and Tom stands behind Mike and he starts patting his back. Very feel-good like moment.
Followed by, or may have been before(?).. Tom getting a little worried about the situation and he shouts at little at Santino - "Sollozzo might not even be in the car Sonny!", in a very big brother kind of way.
Don't remember if those were mentioned but this huge favourite was. The scene between good ole' Pete and Mike talking about when he was fighting for their country in the war and he makes it clear that they were all proud of him.