In the beginning of their relationship, Jack tells Rose that they will go to an amusement park and then ride horses down a beach "but none of that sidesaddle stuff." At the end of the film, one of Rose's pictures that we see is her sitting on a horse, one leg on each side, on a beach, with an amusement park in the background.
The father/daughter relationship that Rose and Thomas Andrews seem to develop throughout the movie is actually quite sweet. Even more so when you consider the fact that Rose's real father was probably a Jerkass who left his wife and daughter with a mountain of debt.
The band has finished playing its last song, and three members finally decide to go...and then the remaining member starts playing one last song, and the other three come back and join in. Also, when the captain turns down a life vest and goes to the wheel to go down with the ship. Finally, when Mr. Andrews, who designed the ship, decides the same, telling Rose "I'm sorry I couldn't build you a stronger ship."
Actually, the maid (Ellen Bird) had only been hired a few days before boarding the Titanic. It is still heartwarming though. Particularly this bit: Ida gave Ellen her fur coat saying she "wouldn't be needing it anymore." After being rescued, Ellen tried to return the coat to the family only to be told by the Straus' oldest daughter that Ida had given to her, and she should keep it.
One scene doubles as a Funny Moment: After Rose leaves her lifeboat to stay with Jack, he embraces her and exclaims "Rose! You're so stupid!"