As I wended the shores I know, As I walk'd where the ripples continually wash you Paumanok, (2-3)
The poem opens in the speaker's hometown. He's wandering the shores that he knows well, just like the waves continually visit the place they know well. There's nothing like going home when you are having doubts about your identity. Perhaps that's why Whitman sets the first section of the poem, which goes on to ask the big questions, in a familiar place.
Paumanok there and then as I thought the old thought of likenesses, (14)
The "old thought of likenesses" sounds like the type of memory that comes around again and again. As the speaker wanders Paumanok, he finds himself confronting the two sides of his identity. Perhaps being confronted with old memories of who he was in the past is one reason he begins to think about who he is in the present.
As I wended the shores I know, As I walk'd with that electric self seeking types. (16-17)
The speaker repeats the second line of the poem, but this time, he isn't alone. He walks along the shore with other "electric self seeking types" who are also in his hometown. Is he saying that all poets must revisit the place of their birth at one time or another? The poets join him in his memory-seeking quest.
I too Paumanok, I too have bubbled up, floated the measureless float, and been wash'd on your shores, (41-42)
The speaker now identifies with Paumanok, the place of his birth. He, like a seaside town, has been living at the whim of nature. Storms in his life wash things to the shore, and the tides of his life wash them away. He sees himself in nature, and it affects the way he looks at places from his past.