What is the correct way to say "where are you going?"?
Also, can someone please explain that の in "あなたはどこに行くの"?
The are both correct. It's just two different expression of politeness.
The first is polite (丁寧語). It is the form that usually ends with ます です.
The latter is less polite and can be used between friends. It uses the form that usually exists in dictionary to end sentences. (What should be form be called? In Japanese it may be called くだけた言い方)
And there is a note on the above page:
which says that it is usually used in plain text (that does not use ます です).
Both ways are correct, but there are lots of ways of saying "where are you going" depending on politeness levels etc. You can also replace the
In Japanese there isn't any distinction between "where are you going", "where are you going to go", "where do you go", "where will you go", "where will you be going" etc, and it's usually determined by context which is meant.
This is by no means a complete list and the explanations may need to be corrected, but I hope it will give some idea of the different ways of saying it:
Both works, although it'd be better if