Whitman states what he's going to do in the poem: celebrate himself. This practice might seem a little arrogant, but we'll just go with it. (It turns out, that he's celebrating not only himself, but all of humanity.)
He lays out some of his ground rules: we're going to believe ("assume") whatever he believes. At another level, we're going to "take on" whatever roles or personalities the speaker takes on. (This is another definition of the word "assume.")
Whitman must have learned to share as a tyke in the sandbox: he offers up the atoms of his body as our own.
He introduces another character: his "soul." In this poem, the speaker and his soul are two slightly different things.
(Just a note: we normally don't call the speaker of the poem by the poet's name, but in this poem, it just makes things simpler, especially since the speaker tells us that his name is Walt Whitman.)
So, Whitman hangs out with his soul, and they look at a blade of summer grass. (The title of the poetry collection to which this poem belongs is Leaves of Grass.)
Whitman describes the air as perfume and says he could get drunk on it, but he won't let himself.
He wants to get naked and go to the riverbank. He is in love with the air.
If you think these images sound kind of erotic, just you wait. There's a reason why Whitman was considered scandalous in his day.