"We're going to visit our aunt for the weekend." Using "the" can be tricky, because you have to consider three things: specificity, count, and set expressions. So, until you master these three topics, you probably always feel unsure as to when to use "the." In your example, it is also necessary to use "for" instead of "at," because "the weekend" is neither a location, nor an hour. Weekend is a countable noun, and, in this case, it is specific. "For" in this case shows purpose and also a set duration of time with a verb of motion: go + location + for + time. I'm going there for two days. I hope this helps, Greg

You don't use 'the' at all in this case. You say, "I'm going to visit my aunt THIS (or NEXT) weekend". You use THE in examples like this: I'm going to visit my aunt THE weekend after next. THE weekend (instead of 'during the week') is most convenient for me to go to the movies. I spent THE weekend cleaning the house.

Yes. It is correct to say both "Are you going away for the weekend?" and "We're going to visit our aunt for the weekend." It is also correct to say "We're going to visit my aunt AT THE weekend" in British English. It is not standard usage to omit the 'the', although this form does exist in some non-standard UK dialects.

Well then I stand corrected because I didn't know it was used in British English. Good to know... I learned something also!

If you're interested in British English, Fraser's answer covers everything. If you're interested in American English go with the answers below. To reiterate once more, saying you are going to do something 'at the weekend' is correct in British English and used by people throughout the UK and Ireland.