Which of the following sentences is grammatically correct?

The grammatically correct sentence is found in the second option because it properly employs the rule of subject pronouns in English. When referring to oneself in a subject position, the correct pronoun to use is "I," rather than "me." Additionally, it is a convention of English grammar to place the other person's name before one's own in polite speech. Hence, "Michelle and I" appropriately uses the preferred structure and pronoun.

In contrast, the first option employs "Me" at the beginning of the sentence, which is incorrect since "me" is an object pronoun; it should not be used as the subject of a sentence. The third option is nonsensical as it combines elements in a confusing way that detracts from understanding who met whom. The fourth option, while grammatically correct, appears more casual and does not incorporate the other person's name in a way that emphasizes politeness and formality in conversation. Therefore, the second sentence stands out as the most appropriate and correct choice.