Is it correct to put 先 before 要 ("我们先要去商店...")?
Yes. I use 先要去 a lot more often than 要先去. 先要去 is more conversational, while 要先去 is more formal, but both get the meaning across.
Funny, I normally use 先 after 要去 in my conversations so 我们要去商店先, 然后回家 should also be correct
我们先要去商店，然后回家 works grammatically as well
It is accepted now.
我们先要去 is acceptable
I literally put the same thing as the translation and it was marked incorrect -_-