the most familiar forms of artwork we see in pictures of ancient Byzantium is its mosaics depicting holy men standing in lines, whom Yeats describes as sages standing in God's holy fire / As in the gold mosaic of a wall..." Most of the sages in these mosaics are depicted against a background of brilliant gold mosaic tiles, which to Yeats suggest "God's holy fire." Evidently the poetry Yeats intends to write, including "Sailing to Byzantium," are to be a form of religious worship.

It is possible to escape from worldly cares and lose one's self in artistic creation. This is what Yeats is actually doing and how he plans to spend the rest of eternity. He may be a genius and a great poet, but his feelings are not much different from those of most men when they grow old. They would like to have something to do with their time. They would like to forget about the fact that they will soon have to die. They would like to forget their weak and tired bodies and the wrinkled faces they have to look at in the mirror each morning. They would like to feel that they are still useful to the world and are not just taking up space. 

"Sailing to Byzantium" bears a strong resemblance to "Crossing the Bar" by Alfred, Lord Tennyson, written when Tennyson was eighty years old and had only a few more years left to live. "Sailing to Byzantium" was written in 1926, when Yeats was sixty or sixty-one, entering a decade which has been described as "the youth of old age."

I believe that this question might be asking about a central theme to the poem rather than a thesis; however, I will try to answer based on the poem having a supposed thesis. A thesis is an argumentative statement, and it is possible to consider that the poem's opening sentence is its thesis statement.

That is no country for old men.

Readers do not know what country the speaker is referring to; however, we do know that the speaker's argument is that it is not an ideal place for anybody who happens to be getting old. The narrator then provides some supporting evidence for the thesis. The country that he is speaking about is a place for the young and in love. People in this country live in the moment and do not think about things that might last for a very long time; therefore, the speaker tells readers what his solution to his age and country problem was. He sailed to Byzantium.

And therefore I have sailed the seas and come
To the holy city of Byzantium.

"Sailing to Byzantium" by William Butler Yeats is a poem, not an argumentative essay. Although argumentative essays by their nature have theses, most poems do not have arguments per se nor do they exist primarily to advance a single thesis. Instead, poems tend to have themes, clusters of associated images, emotions, and ideas that are central to the reader's experience.

The main themes of the poem are the relationships of the soul to the body and youth to age. The narrator begins by suggesting that much of the world is more fit for the young than the old; the young are caught up in romance and the life of the senses and pay little attention to the profound wisdom that Yeats associates with both age and agelessness.

The poem concludes by suggesting that while age involves physical decline, this should not be an excuse for a concomitant mental decline, but instead the soul can continue in the path of wisdom and creation, and participate in eternal beauty. The goal of humanity is to become an immortal creative spirit, casting off imperfect mortal flesh. 

Sailing to Byzantium is a poem by Yeats. In it, Yeats describes what it is like to grow old. The first line states that the country in which he currently lives “is no country for old men”  but rather a place where:

   Whatever is begotten, born, and dies.
   Caught in that sensual music all neglect
    Monuments of unaging intellect

There was recently an award-winning movie whose title was based on this line. The entire poem is a metaphor for a spiritual journey that the speaker embarks upon to the imaginary land of Byzantium:

And therefore I have sailed the seas and come
To the holy city of Byzantium

He has sailed to this city because there, he can be taken out of the natural world where his body is decaying and be transported to a more spiritual world. Most scholars believe that this poem is not so much about another world as it is about the beauty of poetry that is so profound that it is able to transport one’s soul to another world, in a figurative sense.

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.

IV
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.

In William Butler Yeats's poem "Sailing to Byzantium," the place of Byzantium holds the meaning of metaphorical immortality, intellectualism, and artistic pursuit. The speaker in the poem laments the passing of time and the aging of the human body and ponders how one can reach a figurative kind of immortality through leaving artistic and intellectual achievements behind after one dies.

The speaker sees the intellectual and artistic pursuits that occur are integral to the culture of Byzantium, and he dreams of traveling to this land as an old person whose mortal life is beginning to noticeably come to an end. Byzantium is a hope of humans that our short lives will be remembered, and thus become more meaningful, through the lasting impressions and contributions we can leave behind.

Though it is of course a matter of interpretation, if you follow the general bent of the poem, it would appear that Byzantium is a place where age is not as important, where things are permanent and lasting rather than fleeting.

He makes it clear that his current place is not one that really celebrates his age and any old man for that matter.  So he longs for a place where things that are old might be taken seriously.

So in this case, he names Byzantium as a place where he might have the opportunity to live in the world where people appreciate things of excellence regardless of age.  He might even take on a new form there, in order to fit in.

There are many symbols in this poem that you could choose, but the city of Byzantium itself is a good one. This city represents something magical in the poem. The poet travels there because he is old and sickly and is hoping that  this city, which contains timeless art, will revive him. The poet wants to transform himself from a mortal man whose days are numbered to a spirit that can continue in the eternal world of art's beauty. He wants to transcend his mortal body and become one with art. In so doing, he would become immortal. He believes that art is powerful enough to accomplish this, and that the best art is contained in this magical city of Byzantium.

Others will probably respond and give you some other ideas. There are LOTS of symbols in this work. You can read about some of them yourself at the link below.