The title asks you to consider that you don't really know what a blackbird is until you've considered it from different angles. And even then you still only have a limited view of it. There is no one "real" or "true" blackbird.
I was of three minds, Like a tree In which there are three blackbirds. (Section II)
Stevens turns the blackbird into an element in a simile. How is a blackbird like a "mind"? At this point, it's good to remember what Stevens himself said about the poems, "This group of poems is not meant to be a collection of epigrams or of ideas, but of sensations" (source). So, what does a blackbird in your mind feel like?
A man and a woman and a blackbird Are one. (Section IV)
The speaker seems to be saying, "When you think about it, we're all just slices of the same pie. You, me, that blackbird over there…"
But I know, too, That the blackbird is involved In what I know. (Section VIII)
This guy has birds on the brain. Everything he looks at is colored in some way by encounters with the blackbird. We wouldn't be surprised to open up his brain and find a blackbird pulling the strings like a puppet-master.
Once, a fear pierced him, In that he mistook The shadow of his equipage For blackbirds. (Section XI)
Stevens uses deliberately old-fashioned words and images in this section, adding a new layer of reality to the poem: historical time-travel! This is another section that needs to be felt, not interpreted.