I'm loath to comment lest I get shot down but here goes... (just my opinions)
Quickie: I think Internal Affairs is a good movie that most people will enjoy.
Greenaway, like many auteurs, has done some great and less than great stuff. He's very much a 'film as art' -type director and so allowed to make mistakes as well as quite outstanding experimental stuff. And, a bit like the great Mr Lynch, can be brilliant or a bit tedious. Drowning by Numbers is sufficiently mainstream to be enjoyed by the large number of people at a pinch, and Cook Thief is at very least a film you will never forget, even if you didn't like it.
Pillow Book is a fantastic idea and really introduces the tattoo as high art, but I did personally feel it dragged a little. I disliked his Prospero's Books. Shakespeare was one of those gifted artists that can simultaneously write for the plebian audience as well as the very learned, and Greenaway in my opinion failed to understand the wonderful depth of Shakespeare and the specific way The Tempest (on which it is based) links into mythology and psychology. Arty it might have been, though for me Greenaway failed in his duty as an artist respecting a greater artist.
Tulse Luper was an ambitious project, and probably better to know what you are getting into before setting aside time for the full trilogy. I think I would recognize the artistry if I had to review it, but it wasn't personally to my taste. I think I've seen Architect, Zed, and Draughtsman, but I can't remember what they were about now so they couldn't have made a deep impression (although I think they were ok).
If I watched one of his films again, or bought one, it would probably be By Numbers as it can be fun. I've seen Cook, Thief twice, otherwise I'd watch that again (it was worth it to see what subtleties I missed the first time).
