This is a slightly edited and trimmed down version of a post I made on my tumblr.
I saw a lot of people watch House this past October. If you're one of those people reading this, hi! Whenever I find out a friend has seen House I excitedly barge into their DMs to see if they've watched anything else by him and often they haven't heard of any other movies that he's made. So I'm going to share the links I throw on people when they want more information about Obayashi, his career and where to watch his movies.
Firstly I'm not going to link you to a "Where to Start with Obayashi" article, all of them are kind of weird to varying degrees so I'm going to link two career overviews that I think are quite good instead! The first is from the Senses of Cinema Great Director series which mixes biography and filmography. The second is this really good video essay from Pyramid Inu. In it they watch 26 movies by Obayashi and talk a little about each in turn. This one will give you a really good sense of the visual style and storytelling of each.
After this you might ask, where can I watch his movies?
A very kind and diligent twitter user by the name of capybaroness (shout outs) has gone to the effort of putting together a very comprehensive mega folder (which can be found here) which acts as a sort of career overview. There are some absences which are detailed in the FAQ. I downloaded basically everything from there over the course of about a month but I think you can skirt the mega download limits if you use a VPN in my understanding (i think she goes through this in the FAQ also)
Lastly if you are looking for a personal recommendation, I would recommend Bound for the Fields, the Mountains, and the Seacoast (all of the images in this article are from that movie). It's a story about rising nationalism in the months leading up to World War 2 and the way in which kids internalize the things they see adults do around them through play. It sounds like a heavy movie but it's not! until it is, but before that it's a live action cartoon. Incredibly funny and sweet and then the ending completely breaks your heart. If you're going to watch this one, I'd recommend the colour version first because they do something different with the colour version that I can't imagine the black and white version topping. The colour version can be found here and is also available to watch on youtube!
also, if you're sold and have a physical media player, third window has an excellent (but still incomplete! please release more!) selection of his work available on blu-ray.