Does the anime series of Bleach stick to the manga's plot? Or is it like the original Fullmetal Alchemist series where the anime is its own thing and doesn't follow the story of the manga?
The anime pretty much stick to the manga, except of course during the filler arcs. The filler arcs are the Bounto Arc, Third Squad New Captain Amagai Shuusuke, Zanpakutou Rebellion, and the Gotei 13 Invading Army arc. There are also occasionally filler episodes among the canon arcs. A detailed comparison is available on Bleach Wiki.
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For the most part, the anime does stick to the content found in the manga. It does have a few filler arcs here and there and some slight changes, but for the most part it is the same. (For specific comparison episode-by-episode, you can head to the Bleach Wiki here.)
To cite your question, the reason why the Bleach anime sticks better to the manga than the original Fullmetal Alchemist is because of timing. The original Fullmetal Alchemist anime started shortly after the manga, and since it was faster to turn precreated content into a new format than it was to create new content, the anime quickly outpaced the manga and thus they had to start making up their own material in order to keep producing the show. The Bleach manga on the other hand, for a while it stayed about level with the anime. But at the end of the The Lost Substitute Shinigami arc (episode 366/chapters 476-479) the anime stopped airing (for more information on why, look at a previously asked question here). So as of that point, there is no more anime to stay with the manga. But the manga is still going, according to mangaupdates chapter 635 was just released on 7/15/15 by means of MangaStream.
So in review, yes. For the most part the anime does follow the manga... until the anime stopped.