Ultimately, different clubs here is just a colloquialism for various clubs. But different is actually not needed, since you can just say "I go out to clubs with my boyfriend" and everyone will know you mean various or more than one. As is, it is functioning as an indefinite article.
Adding a or the in front of different would be modifying clubs. To see what they would mean on their own, let's try the phrase without different.
I go out to a clubs with my boyfriend.
This is grammatically incorrect, since a clubs doesn't agree in number. You could use some clubs to make the number match. But adding back in different is then confusing, because "I go out to some different clubs with my boyfriend" stops different from functioning as an indirect article, suggesting an intentional contrast not heard in the original statement: perhaps it means clubs different from some particular one, and the boyfriend is a different person than present company or a known/understood person.
I go out to the clubs with my boyfriend.
This could be correct, speaking of a definite set of clubs known to the listener by inference or previous reference. But the moment you add back different, you get a conflict in meaning. It makes no sense to use a definite article with an adjective that itself strongly connotes no particular set of clubs. The different sounds extremely unusual. You could say the other clubs but you'd be referring to a particular set of clubs, either itemized or, in contrast to a particular club, all the ones that aren't that particular one.
I go out to a different club with my boyfriend.
This sentence parses correctly but has a narrow meaning: it implies that in comparison to a club known to or understood by the hearer—such as the current club if spoken there—there is exactly one particular club you "go out to" with your boyfriend ("a different" one). If instead you mean multiple clubs or clubs in general, then the new sentence does not work.
If you would explain carefully the situational context, and the exact meaning you intend, it would be easier to tell you the correct sentence to convey that meaning, and it would be easier to explain why all the other options mentioned don't do the job.