In your sentences, only "He joined a club with me" is correct. In your sentence, "joined" is a transitive verbs and needs an object; "a club" is the object. Normally, an adverb or adverbial phrase does not come between the verb and its object, so "with me" could not be used in the position between "join" and "the club." _______
"Join with" has a meaning very similar to that of "join." "Join" means to become a part of or an addition to. "Join with" stresses a more cooperative effort, as in these examples from Google:
All of the instances of "join with" in the sentences above could be replaced with "join." Some writers think that "join with" and "join together" are redundant phrases and that "join" is enough. However, "join with" does give the impression of more interaction.
"Join with" also carries the idea of "for the purpose of." In your sentence, you "join the club." You have not stated any purpose.
However, you might say: "We joined/ joined with the club in order to strengthen our position."
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