This poem is so complex that several approaches can be taken all tending toward adding additional understanding of the meaning of the poem. In other words, the more Answers, the merrier!
Your question asks for "surface meaning" and "deep meaning." These are technical terms relevant to Formalist critical analysis. Surface meaning equates to what we call "plot"; the Formalist term for this is sjuzhet and defines the particulars of a narrative that are unique to the work being considered. Deep meaning equates to what we call "story"; the Formalist term for this is fabula and defines the universal elements of a narrative that may be found (indeed, for Formalists, is found) in myriad works.
To be specific, here, the surface meaning is the fantasy narrative of an old man metaphorically sailing in imagination to ancient Byzantium to meet with the sages of old, whose art lives on after them, from whom he wishes to partake of the blessing of living art. The deep meaning here is the quest for the True Muse, the True Fount of True, Living Art that effects the foundations of souls and of human life.
Briefly, the sjuzhet (surface meaning) is expressed by the poetic persona (Yeats, himself, in this lyric in a modified octava rima) who wishes to transcend the land of lovers and perpetual summer,
at their song,
The salmon-falls, the mackerel-crowded seas,
Fish, flesh, or fowl commend all summer long
where there is no hope of learning to "sing," remembering that "song" and "sing" are conventional metaphors for a poem (song) and for the contribution of a poet (sing). He desires to gain admittance to a higher plane of consciousness for which he has set course: "the holy city of Byzantium."
Once there he beseeches the wise sages of ancient Art and Beauty to be "the singing-masters of" his soul to teach him true poetry and the source of true poetic expression. The final lines are subject to various analyses, but one analysis is that the poetic persona expresses his desire to transcend the lowly, base physical realm and be forever embraced in the artistic eternal so that he might escape the physical:
... gather me
Into the artifice of eternity.
Once out of nature I shall never take
My bodily form from any natural thing,....
In other words, he would prefer to be in the shape of a mechanically crafted and enameled nightingale of old that sang to an Emperor rather than to be in the shape of "bodily form" that knows not True Art. Of course this is all metaphorical, and Yeats is not expressing a wish for real physical death. On the contrary, he is expressing the fabula (deep meaning) of the quest for the True Muse, the True Fount of True, Living Art, which of logical necessity can only be expressed in life by someone who is living.
Yeats, here in his old age (1928), is reflecting on the rapid movement of life, from youth and fecundity to old age and impotent senility. The images of fowl and fish represents youthful fecundity, and the scarecrow images (“a tattered coat upon a stick”) represent his present aged stage. He argues that “song” is the best weapon against aging; for Yeats that means writing poems that advocate a free Ireland. He retreats to “Byzantium” (Constantinople) (by which is meant exotic religion and philosophy), speculating about the afterlife, in which he shall take on the figure of a piece of art rather than “any natural thing.” The poem is heavy with Jungian universal symbols (“golden bough”), as well as personal subjective images from Yeats’ own experiences (“gyres”, for instance, for Yeats, refer to the ever complex worldview that threatens to break apart the traditional order of Man’s world). It is probably best to approach Yeats from his more accessible early work rather than the sophisticated work of his later life This poem is complex, combining his spiritual inquiry with his reflections on the meaning of life.