The Last Samurai. I know what you're thinking. How can Tom Cruise be the last samurai? He somehow pulls it off. This is actually one of my absolute favorite martial arts movies because it says a lot about a very strange time in history for Japan, and delves deeply into the philosophy of bushido and what it means to be a warrior.
Return of the Dragon. Really, any Bruce Lee movie (except Game of Death, because it wasn't actually Bruce Lee for 80% of the movie) could make this list, but Return of the Dragon holds a special place in my heart because it was the first Bruce Lee movie I saw. Also I think that if one were to share a Bruce Lee movie with their family, this would be the most appropriate. It has a relatively simple but compelling storyline and it's probably the least violent Bruce Lee movie.
Kung Fu Hustle presents itself as a goofy martial arts comedy, and while it does deliver laughs in almost every scene, there is also a lot to be said about it. The martial artists in this movie are just everyday people who want to avoid trouble as much as possible. It isn't until they absolutely have to that they even reveal that they're martial artists. Despite the over the top laughs, this movie actually paints a pretty accurate picture of the martial artist as an everyday person who just wants to live peacefully.
Drunken Master is a comedic imagining of the early years of the legendary Kung Fu master and doctor Wong Fei Hung. The story takes him from being the class clown in his father's dojo, to becoming a young master who returns home to his father a changed man. As you can guess from the title, Wong is sent to be trained by his uncle in the Kung Fu style known as Drunken Boxing (which in reality doesn't involve drinking, it just imitates the off-balance movements associated with it). This movie also has a lot to say about the master/student relationship. Wong hates his teacher at first, but grows to see him almost as a second father figure.
Once Upon a Time in China actually follows the life of Wong Fei Hung as well, but this movie series is about when he is already an established master martial artist and doctor. This is much closer to what historians say about Wong Fei Hung. While there is some comedy sprinkled throughout the movie it is a much more serious movie than the previous one.
*Note* I'm not endorsing these movies for people of any age. I do recommend that parents look up these movies on imdb.com which has a great feature that lists anything that can be offensive/inappropriate in a movie right down to the exact number of swear words.