Translation should be 'Do you know where we’re going TO?
I think it’s quite common to skip the final “to”, though I suppose most purists would prefer the variant you’ve suggested.
By the way, it’s actually quite similar in Polish: the hypercorrect usage in sentences like this one is “dokąd” (=“where to”), but many Poles just say “gdzie” (=“where”) in casual speech.
Thanks for that alukasiak. I did wonder if ‘gdzie’ would also fit into this sentence.