I love the part when Jack and Fabrizio win the Titanic tickets and rush onto the ship. Their happiness just kind of spills onto you as you see them waving good-bye and then just marveling at the speed and size of this ship. You notice that you don't need to be rich to truly enjoy yourself and have a great time. Everyone is excited about this boat, and you become excited and exhilerated as well.
This is one of the best happy parts of the movie. This scene was original, creative, and extremely romantic. And, come on girls, you had to admit that you wished you were in Leo's arms at that moment. I guess it was just the way that whole part looked that really got to me.
The Steerage Party
The party was really great. For one of the first times in the movie, you see Rose as herself and not some trying-to-be-proper rich girl.
Rose Dies and Meets Up With Jack
Yeah, I know I have this scene included in my crying section, but I believe it is a happy part, too. You see that Jack and Rose can be together again, and you take notice to people that have died on the ship.
The way the sunset framed Jack and Rose, the color of the water as they looked upon it, and just the colors of the ship really suck you in and make you enjoy the flying scene even better. I also love it when the camera goes away from a close-up of the couple and shows the full bow of the ship. The best part is that the sunset is actually real! 
Jack did a perfect job coaxing the real Rose out. "I don't know the steps!" Rose had said. "Neither do I, just go with it!" was his reply. The music is upbeat, the dancing is active and you just want to get up and dance yourself. 
You get this reassuring feeling that these people are living on in a better place and will actually never ever die as long as they are remembered.