Much of what I know about Chan is based on a reading (now many years past) of his book I Am Jackie Chan: My Life in Action (1998). And watching more than half of his huge movie output, some of them multiple times.
Jackie Chan spent his childhood in a Chinese Opera school, run by a master that treated the children like indentured slaves (my term, not Chan's). The students worked and trained 16 hours a day (I'm going from memory here, and I admit it's been a long time since I read the book). They learned to dance, sing, and do acrobatics. What I remember most from the book was his extremely adversarial relationship with one of his "big brothers," and his protective stance toward one of his "little brothers" - "brother" meaning other students raised in hardship at the school. He really hated that big brother - a guy by the name of Sammo Hung. And he did his best to take care of Yuen Biao. All three of them went on to work as stunt men in Hong Kong cinema, and as soon as they stepped outside the school their relationship changed radically. Sammo had been the school bully, but on the outside he saw it as his duty to take care of all his little brothers - and he helped both Jackie and Yuen get started in the Hong Kong film industry. Chinese Opera was on the wane, but the HK film industry was on the rise and in dire need of stunt men - for which the brothers' acrobatic training made them eminently suitable. Chan worked initially in any production that would have him - leaping, falling, and dying in 15 second walk-on parts. He was eventually noticed, and started to land bigger roles - initially in other company's movies, but eventually in films where he had creative control.
When he was starting out in the 1970s, martial arts movies were the exclusive domain of Hong Kong and tended to be extremely violent with lots of spurting blood and revenge and a complete lack of humour. Chan dominated the market through the Eighties and Nineties, with his preference for humour (as cheesy as it often was) and generally bloodless fights. He was also unusual in that his characters tired as they fought, and showed pain. I rarely find Chan's "humour" amusing, and his idea of chivalry frequently stumbles into sexism, but I've always preferred his emphasis on athleticism over maiming. Chan became particularly well known for his use of props, turning almost any object into an obstacle, jungle gym, or weapon.
His "brothers" are an interesting comparison: Sammo has always been a bulky guy, but he remained shockingly agile into his fifties, having a very long and successful career in HK (and to a lesser extent in North America - anyone remember the TV series "Martial Law?"). Yuen Biao was more agile than Jackie - sadly, he didn't have the charisma of the other two and the few starring turns he had were in low budget and not very good movies (I've seen a couple of them).
Caveat
Realize that the measuring scale I use for martial arts movies is radically different than the one I use for regular movies: martial artists are rarely good actors (Chan certainly isn't) and the plots are primarily about "how can we fit in more fights?" rather than character development. As such, I judge these movies on the quality of the fights and stunts much more than the acting or plot. In the case of Chan's movies, I find some of his "humour" so awful that I have to fight an urge to downgrade movies with great stunts and fights because of the cringe-inducing jokes ...
Best Fights
Sometimes cited as the best martial arts fight ever put on film (definitely by me, but also by others), the fight between Chan and Benny "The Jet" Urquidez toward the end of "Wheels on Meals" [sic] is both impressively long and truly spectacular in its speed and inventiveness. The plot and humour of the movie are crap, but it contains many superb stunts and fights. Another favourite of mine is the roof fight near the end of "Who Am I" in which Chan takes on two very good martial artists (both real-world students of his). This one is played even more for humour, but is also spectacularly athletic. Again, a silly, sexist movie full of great stunts and fights. In "First Strike," he uses a three metre long step ladder as a weapon - very effectively (one of his better movies overall). The final fight in "The Drunken Master II" also rates as one of his best. "Dragons Forever" has all three "brothers" plus Benny Urquidez and some excellent fights (and if you doubt my comments about Biao's agility, watch him in this one or "Wheels on Meals"). I should probably also mention the playground fight in "Police Story 2."
Best Stunts
Chan has, at one time or another, broken nearly every bone in his body. His best known stunts include sliding 21 stories down a 45 degree slope in "Who Am I," and falling four storeys from a clock tower through a couple canopies to the ground in one of the "Project A" movies (complete with a tribute to Harold Lloyd's iconic clock scene). Other particularly famous stunts include dangling off a careering double decker bus by the crook of an umbrella (one of the "Police Story" movies), driving a car downhill through most of the buildings in a large shanty town (also a "Police Story" movie), and sliding down a four storey pole through electrical wires (another "Police Story"). But I often favour the smaller tricks he does: in "Who Am I," trapped in a boardroom with someone coming in, he "vanishes" by leaping against a wall to a picture frame to a closing skylight and out in about a second and a half - levitating approximately four metres straight up in the process. In "Mr. Nice Guy," he rolls over a live table saw blade - humping his body up at the right moment to not be sliced in half. He does these things with a casual ease that's awe inspiring. The truth of these stunts is more complex: he's said in interviews "[anyone] can do these things," and proceeds to explain that all you have to do is practice, and be willing to commit to the 100 or so takes that some of those stunts required.
For those not familiar with his cinematic convictions: until around the year 2000, not a single one of his movies included wire-work or speeding-up. This is in direct contrast to a huge portion of the HK martial arts output, but Chan wanted his movies to look realistic and it truly pays off. I should mention one caveat to this: he did use wires in some shots for safety, but if the wire came into play - that wasn't the shot that ended up in the movie.
Props
His use of a three metre step ladder again comes to mind as one of his best props as well as an excellent fight. But his movies (after 1980) are always loaded with props. One of my favourites is again in "Mr. Nice Guy," in which he stuffs someone in a concrete mixer (and turns it on), zip-ties someone else to a soil compactor (and turns it on so they're bouncing up and down), and (as mentioned) rolls over a live table saw. I should mention that "Mr. Nice Guy" is an atrociously bad movie even by the low bar set by Jackie Chan's movies, but it does contain some fairly awesome stunts and fights.
Film History
Of his early movies (still in the 1970s), I recommend "Snake in Eagle's Shadow" and "The Drunken Master." These are both far enough along in his career that they feature his humour and focus somewhat less on bloody revenge than standard HK martial arts movies of the period, but both are relatively traditional (if more inventive) in their fight filming. In the 80s he got away from the traditional style and moved into films set in the modern day, and started using anything-as-a-weapon. As an entry point to his movies - particularly for those not familiar with the HK aesthetic and/or Chan's quirky sense of humour - I would most recommend "Rush Hour." I haven't listed it among his "Best Stunts" for a reason, but it still has uniformly good action and humour that translates better to North America than most of his other movies and many good fights and stunts (handcuffed-to-a-steering-wheel comes to mind).
As Chan got older, he became less acrobatic and his movies tended to lean toward seeing how much pain he could absorb. Possibly the best example of this in North America is "Shanghai Noon," with Chan falling about four stories down the inside of a clock tower, smashing into things as he went (reminiscent of the famous "Project A" tower fall, now that I think about it). Later still, his movies show poorer, less agile or athletic stunts than his old output - although he returned to self-abuse in the awful "Skiptrace" with Johnny Knoxville.
In his later years, he tried to encourage a new generation of Kung Fu action movie stars by putting his money and relatively short appearances into movies like "New Police Story" and a couple others. "New Police Story" did fine at the box office, but I don't think it launched any careers.
Acting
As I've mentioned, Jackie Chan isn't a particularly good actor. "The Karate Kid" showed him doing a decent if not outstanding job - something he should have managed 20 years prior had he actually been an actor. I've always considered "The Foreigner" an interesting case. Chan stopped being a sixty-year-old trying to pretend he was thirty five again, and instead embraced his age. This was who his character was. Instead of being an old guy who was pretending he was young but was of course unable to move like a young man, in this movie he's an old guy who happens to have an utterly terrifying skill-set. It works far, far better. The movie foregos any of Chan's usual humour for a serious dramatic turn from Chan. His acting, while not Oscar-worthy, is quite good.
His voice acting in the "Kung Fu Panda" series is worth noting: it's enjoyable and he's quite funny, as well as offering a tribute to his very long career in martial arts movies. And the movies themselves are absolutely worth seeing: they are (all three of them!) possibly the best martial arts parody films ever made.
Credits and Outtakes
Chan was particularly famous through the 1980s and into the 1990s for interspersing outtakes into the closing credits, usually failed attempts at stunts or fights. Sometimes these are comedic (somebody falls into someone else while attempting to kick) and occasionally horrific (several instances of Chan being carried away after a serious injury with a stunt gone wrong). It's a fascinating view into his work process. As he got older, these shifted more and more to comedic out-takes - often people mis-speaking lines - with the result that they're much less interesting.
The "Must-See" List
I would most recommend "Rush Hour," "Who Am I," and "Wheels on Meals" (probably in that order if you're new to his output). If you're enjoying it, then you should definitely work through the "Police Story" series, keeping in mind that in North America "Police Story 3" became "Super Cop," and "Police Story 4" is "First Strike."
Chan has been impressively prolific. I'm rarely a fan of his comedy, and most of these have his sense of humour front and centre, but they are also from the period when he was at the height of his physical skills and include incredible fights and jaw-dropping stunts: "Project A," "Armour of God," "Project A Part II," "Armour of God II: Operation Condor," "Drunken Master II," and "Rumble in the Bronx."