is "Where are you going to?" correct
Solution 1:
The “to” in “Where are you going to?” sounds superfluous to me just because “Where are you going?” is perfectly fine and I cannot think of any reason to add “to” to it.
The reason why “Where are you going to?” sounds strange is not because the sentence ends with a preposition. For example, there is nothing wrong with the sentence “Who are you talking to?”
Solution 2:
It's worth checking out the other preposition threads on the site. I especially like
Prepositions at the end of sentence and whom
which states ending on a preposition is acceptable usage, and only frowned on because of an over-exertion of Latin rules by grammarians.
Solution 3:
Both examples are quite acceptable in colloquial speech, though the former avoids the hanging preposition, so is preferable to the prescriptivists. Also, the word "where" has subsumed the dative sense ("to where", or "whither" in formal/archaic English.)
In any case, I would consider three variant forms, which are:
Where are you going [to]? (most common)
To where are you going? (grammatically more "correct", though perhaps the preposition is a bit redundant)
Whither go you? (a bit archaic, but I do like how succinct it is)
P.S. Regarding hanging prepositions, even many men of letters in the past have had no qualms about using them; it seems to be a grammatical rule artificially copied over from Latin at some point.