Americans use #1 (Let's not go) #2 (Don't let's go) is more common with British speakers. Both will understand you. Good luck!

I agree with Dan's except that I think the second is correct. The second one, "do not let us go" makes good sense. It is correct. With the third, I struggle more. If you eliminate "not", you have: "Let's do go" I like the way that sounds, but I am afraid it is wrong. I cannot justify it. "Do" can be followed by "go", as in "we do go", but what makes that work is that "do" is an active verb, not an infinitive. However, for "let do" to work, "do" must be an infinitive. You cannot have it both ways. "Do" can not be both an active verb and an infinitive so the sentence fails.

They all mean the same thing. In US English, "Let's not go" is standard, natural phrasing. It is definitely correct, and it's the one you should use. "Don't let's go" is varying the word order to add a little emphasis or personality. "Let's don't go" is kind of a joke, and would be expected mostly as a humorous retort or reply to "Let's go." I am not an expert but I think the first is grammatically correct and that the other two are not. However, phrases of this kind are idiomatic. They are units in themselves and shouldn't be analyzed grammatically. In any case, all of these phrases combine "let's go" with a third word expressing contradiction or negativity, and are understood and mean the same thing. Changing them from the expected "let's not go" calls attention to them and adds emphasis.