Best Known for the Fanservice: The mizuage scene was so controversial that it's probably the best-known moment of the film (and thus what makes casual observers remember most about geisha, much to the chagrin of real geisha).
Broken Base: The casting of the three Chinese actresses (though Michelle Yeoh is Malaysian with Chinese descent) as Japanese characters sparked furor among the Chinese and Japanese audiences. While the Japanese are upset that no Japanese actress got any of the main roles, the Chinese were very hostile about it which doesn't help that the fragile historical relationship between the two countries since World War II which the movie is set in only fanned more flames. However, several people such as Ken Watanabe defended the cast, citing the actresses' performances was the reason for the casting.
Draco in Leather Pants: Aside from being enemies of Sayuri, Hatsumomo and Pumpkin earned this kind of attention from the movie's audience.
Jerkass Woobie: Pumpkin. Not as pretty as Sayuri, not as talented as Sayuri, not as smart as Sayuri, not as lucky as Sayuri, and browbeaten by her supposed mentor Hatsumomo into the bargain. Oh, and Sayuri is unwittingly responsible for Pumpkin's downfall.
Hatsumomo as well. While she is a sadistic bitch, like Sayuri she was not free to be with the man she loved- and was even harshly slut-shamed for it.
Tear Jerker: Chiyo was sold to an okiya- a very harsh, bitter place where she was abused relentlessly, lost both her parents, and was separated from her sister- whom she never saw again. And that's only the beginning!
Pumpkin's revenge against Sayuri is meant to be a major Kick the Dog, but as she points out to Sayuri, Sayuri took Pumpkin's life goal of becoming officially adopted by the okiya and becoming one of the most successful geisha in Gion, and she didn't really even want that; it was all just "a stepping stone" to get to the Chairman. Under those circumstances, it's easy to feel like Pumpkin's revenge was justified, even if Sayuri hadn't meant to ruin Pumpkin's life.
Similarly to the Pumpkin situation, Nobu's reaction to Sayuri could be seen as extreme, but she did make him believe she loved him, under instruction from Mameha, so his feelings of betrayal when she and the Chairman get together is pretty justified, especially as the Chairman pretty much orchestrated the whole thing.
The Woobie: Chiyo, at least until Mameha decides to take her on.