All three did you go, have you been and had you been seem okay to me. However, I'll try to tell the subtlety I find.
Firstly, I'd select the verb been over go. That's because have/had been refers to go to a place and come back. In the context of visiting a place, been serves better than go. But that does not mean the first question is incorrect. A note for the first question though - when you and the listener are talking about the same spot, it'd take the definite article the.
In an informal speech, you may ask
Did you go to the picnic spot (which) you were asking me?
This means you are asking whether the person went to that place.
Had you been to that picnic spot which you were asking me?
Looks okay (using that instead of the talks about the same spot, so article is omitted). But then using had gives a flair of more past and not recent past.
Have you been to that picnic spot which you were asking me?
Also looks okay. Just using have here I think talks more about the recent activity. It may seem that he has just visited the place and you are asking this. Using have + participle is recent as compared to had + participle. Compare I had done this over I have done this. The latter is recent past or at least its effects has remained as on the time the sentence is spoken. Good to note that in your very own question you used "....I had have visited long back." That supports it!
But as I said, all three seem okay to me.
I agree with Joe Dark's comment: Where is your family originally from?
Additionally, I'll say that a lot depends on the tone of the question. I'm curious about accents, and often ask a person if I can't place his/hers. But knowing that this can be a sensitive issue for some, I often couch the question in some tentativeness:
I hope you don't mind my asking, but I'm where's your accent from?
For what you want to ask, I'd go with something like what you propose:
I hope it's not rude to ask, but you seem to know a lot about Italy; is your family from there?
Asking in advance for pardon and giving a reason to be asking will help, particularly if you're showing a genuine interest.
But in general, I think you're on the right track simply by being aware that this can be a sensitive question.
Best Answer
Maybe you could say it like this:
Repeating words it's not good and you should avoid it as much as possible. But it depends, if you're writing something informal don't be too concerned. If it's formal, or a story, you should try to put different words. A dictionary of synonyms could help a lot.