I'll second the suggestions of the Drunken Master movies, and add Wheels On Meals. Wheels on Meals is a goofy fun movie with some stellar fight scenes and also stars legends Sammo Hung and Yuen Biao. Pretty much any of the Chan Hong Kong pictures of the late 70's and early 80s are worth seeing - even the mediocre ones have some amazing moments.
My favorite Jackie Chan sequence is probably the bicycle sequence in Project A, which is one of the most creative fight sequences ever.
I first encountered him in "Who Am I" and just started renting and netflixing like crazy. Mr. Nice Guy isn't bad, but terrible acting. Police Story is great - I didn't know there were more! First Strike is good as well. OOOO! Project Condor.
Sadly, that was my introduction to Jackie Chan. I vaguely recall him saying something (subtitled) along the lines of "these night vision goggles are the cat's pajamas" - which is where I'm assuming I started using the phrase "cat's pajamas" way too much.
Years ago when my son was small, we were watching some Jackie Chan movie. He turned to me and said, "You know, Mom, if you wanted to get married again - it would be okay with me if you married Jackie Chan. I wouldn't mind if he was my stepfather. Why don't you do that?"
Now for the main reason to watch this movie:
If you ever wanted to see Jackie Chan do Chun-Li from Street Fighters cosplay and fight like her (as well as other Street Fighter characters), this is the movie you should watch. Or, you know, just Youtube that one particular scene or something.
Armor of God is really cool for amazing stunts and kung fu. Police Story 3 aka Supercop is also waaay up there as Michelle Yeoh is is in as well, does her own stunts and is as crazy as Jackie. For a great film starring Yeoh, see Wing Chun, in which she holds her own out numbered 40 to one. And is a professional tofu-balancer.
Drunken Master 1 and 2 are great, and Half a Loaf of Kung Fu is much better than you'd think it is!
Beware really early films, as he only has bit parts in them even though they're billed as Jackie Chan films.