Moments of sadness:
I think McCoy in the scene with his father on TFF was very moving.
I also thought Kirk's scene after the death of his son was moving.
Kira had some incredibly emotional scenes throughout DS9.
And "The Visitor" definitely made me tear up.
Moments of anger:
It was always great to see Worf get angry both in TNG and DS9, because the audience was usually cheering him on!
Loved Kirk's anger at Khaaaan!
Also loved Picard's anger in First Contact.
It was uncomfortable watching O'Brien's anger issues after his experience in the episode "Hard Time," especially when he was around his family.
The whole scene with Spock and Valeris and the forced mind meld that followed was uncomfortable to watch.