Let's try fleshing out a couple of sentences with the starting words in the question and see what's different:
I wish I had gone out before it started raining.
I wish I would have gone out before it started raining.
If we ignore the "I wish" part of the sentence, the difference between the two is a conditional tense in the second one (the "would"). Again, without the "I wish", we'd have the first sentence indicating that the person speaking went out in the past and did so before it started raining, whereas the second sentence indicating that the speaker didn't go out but would have done so had some unspoken condition been met.
But the "I wish" changes the first sentence from a pure past tense sentence ("I did go out") to one where it didn't happen ("I didn't go out") with regrets that it didn't happen ("I wish I had").
The second sentence changes from a conditional one ("I would have gone out") to one where it didn't happen and with regrets that it didn't happen.
And so I think the addition of "I wish" makes the thought being expressed equivalent. That said, the first sentence is grammatically correct and the second sentence suffers from an inappropriate use of the conditional tense.