Which is correct and why? First alternative sounds correct to me, but I can't motivate why.
"Let's don't" is certainly informal, and the grammar is quirky, but I think it's dogmatically prescriptive to say it's incorrect. It's not a truly rare spoken usage either, which suggests it's doing some kind of work that "let's not" isn't doing.
One thought is that it can imply a certain location or context in an ongoing dialog that "let's not" doesn't capture. For example:
A: Shall we go to the park today?
B: Oh, let's do!
C: Let's don't.
Here, C can be read as explicitly regarding and challenging B. As a speech act, "let's don't" seems to have a parameter that the preferred usage lacks. Moreover, this dialog is perfectly intelligible and nothing hits the ear wrong as far as I can tell.
Another thought is that the "let's don't" formulation deliberately conflates the normal structure of a hierarchical injunction with a more egalitarian proposition, either to borrow authority from the injuctive structure, or to minimize the rank of the speaker for whatever reason, e.g. the boss says "Let's don't miss our deadlines, team."
In general, "let's" and "don't" are in tension because one admits the possibility of refusal and the other doesn't. "Let's don't" might either be a "Let's..." statement with heightened emphasis, or a "Don't..." statement with a kind of honorific egalitarianism.
The first is correcter.
"Let's not go there", while a bit arcane, is valid syntax.
"Let's don't go there" is not strictly proper syntax (though it has a bit of an AAVE sound to it). (It's not proper syntax because the triple verb "let us do not go" breaks the basic rules of sentence construction.) The origin, however, is likely from adding "let's" to the correct (in other contexts) "don't go there", a sentence that is perfectly valid English.
Welcome to ELU. For future reference, you need to provide some context for the question, and show what it is you do not understand and why.
That said, your question is interesting. On the face of it, the answer is obvious enough. The first sentence is straightforward enough.
- Let's not go there.
This very current figure of speech is a current way of indicating that a particular topic or suggestion or proposal is in some way unacceptable, risky, dangerous or perhaps offensive. It suggests that a discussion or plan is taking the participants to the conversation to a dangerous place, either because of real physical danger or social, political or reputational risk. The significance of this expression is dependent on what was said before it. It follows standard grammar, provided there has a place (literal or metaphorical) to which not to go. "Let's", of course, is a first person plural imperative, and short for 'let us'
Let's don't go there.
This I find is a reference to the title and first line of a song by Jackyl (which is the information that you could helpfully have included in the question). It sounds wrong. The standard alternative to let's not is don't let's. So Let's don't is not the normal way of saying it. So you could say it is the incorrect one. At least one website, Cummunity Language Tool picks it up as an error, thus proving the present limitations of algorithms.
You could analyse it grammatically. The author has taken the (correct) 'don't go there', which is a correctly formed second person imperative, and turned it into a verb, and the Let's is turned into an injunction for us to "don't go there".
The song is a cry against excessive caution and giving way to the fear of coming last, losing a bet, or being caught in the rain, and the opening and title line holds out the challenge and the promise. Sadly, (I apologise for the opinion) the rest of the lyrics fail to fulfil the promise. Pity