The speaker first asks if he shall compare the person he loves (thee) with a summer’s day. He then does just that. It is almost like he is saying “I will now compare thee to a summer’s day.” By starting the sonnet with a question, he invokes a sense of respect to the person he is speaking about. He may be alone and just speaking rhetorically. But if he’s addressing this person, he is asking permission to make this comparison, knowing she/he will acquiesce.
The speaker then proceeds to describe how his loved one is more lovely and everlasting than the beauty he sees in nature. The speaker never rejects the comparison. He uses the comparison to show his favoritism for his loved one over nature.
In an indirect way, you could say he rejects the comparison because it is through the lines of this sonnet that his loved one lives on. In other words, the eternal summer of his loved one will never fade but that’s because this poem will survive long beyond her death. So, it is the sonnet itself that never fades. The speaker is talking about the beauty of this person, but the comparison between his loved one and a summer’s day is used to make the point that it is the poetry, not the beauty itself, that provides this longer life.
The sonnet opens with the question "Shall I compare thee to a summer’s day?" The subsequent 13 lines do, in fact, compare the beloved to a summer's day; thus, the question is answered affirmatively.
At first, the sonnet appears to be following common Petrarchan conventions of praising the beloved. A summer day is something most people find pleasant, and Shakespeare explores ways in which his beloved is even more delightful than a summer day. He does this by denigrating the summer, arguing that some days in the summer are uncomfortably hot, while others are chilly and rainy (especially in England).
He next argues that, although individual summer days and summer itself are fleeting, the beloved's beauty will last forever. At first, this appears paradoxical, as we know that people grow old and die. The final couplet resolves this by saying that the beloved's beauty will be immortalized in his sonnet and therefore will endure as long as people read.