Both variants are indeed correct, with regard to the
word order, but, as M. Zuberbühler said, the aus is unnecessary, and the sentences are both in fact grammatically incorrect if you keep it, and here's why:
Most likely you're thinking of the way answering is done in English by repeating as much of the question in your answer as possible. (Which I was taught in school showed politeness, shrug; something I feel is unnecessary in German (by which I mean the repetition, of course; politeness is always necessary in conversation). I would be perfectly alright with "Ja, ins Kino" as an answer if I asked you: "Gehen Sie abends oft aus?" - as the asker I know what I asked you, I wouldn't need you to repeat my own question back to me. But that is just my personal preference, and anyway I'm degressing.)
You should however not repeat ausgehen in your answer, nor would you do so in English:
Q: Are you going out tonight?
A: Yes, I'm going to the cinema tonight.
It's the exact same thing in German. Since you're specifying a place you're going to -- the cinema -- you don't need the generic out anymore, in fact it feels wrong to keep it. "Yes, I'm going out to the cinema tonight.", sounds wrong in English as well (to me at least, but I'm no native speaker).
So the use of ausgehen is correct if it is the translation of a generic going out, but use ins (=in das) Kino, Theater, Cafe, Restaurant / in die Stadt, Schule, Disco gehen if you specify a place or places you're going to, leaving out the "aus", as you would leave out the out in English.