Bukowski, unlike most poets, didn't actually select poems for his published books himself. He sent everything he wrote to his publisher and his publisher threw them into volumes. As a result, the poetry collections are much less cohesive than an average book of poetry. The collections also don't exhibit continuity through a given volume (i.e., each poem generally stands alone and is meant to be read as such). Because of this I don't really think that reading any given book would be too different from just looking up poems on the web and reading them at your leisure, if you wanted to go the cheap-and-easy way. - fwiw - I believe I read the bit about the editor in an article about Bukowski, but don't see it mentioned on Wiki so like, 20% chance I'm making it up/confused.
Beyond that, I personally feel that his earlier work is better, so I would recommend earlier books over later books as a general rule of thumb. "Burning in Water, Drowning in Flame" is very well known. My first introduction to his books was "You Get So Alone At Times That It Just Makes Sense."
If you live near northern Delaware, the university library has some really cool Bukowski items. I remember they have something like a small chapbook that he put out - I think it's mimeographed or something - but it's really cool, because it's much more personal and rare than a published book. You get a different look.
He also wrote prose but I haven't tried that.