As far as I can see the asker wants a rewrite of "They have decided to go to a week after next week." We don't do rewrites, it comes under proofreading.
Also, the user doesn't seem to have a strong enough grasp of English to make them a good fit for the site. We are for "linguists, etymologists, and (serious) English language enthusiasts" after all.
I support Matt Ellen's answer and have this to add:
First off, the OP gives a perfectly fine example of an English phrase "a week after next week," but without explaining why he doesn't just use this phrase, gives three alternatives and doesn't describe why he is confused about the choices he gives.
I voted to close as Not A Real Question, because it is difficult to determine what is being asked. Now it may be that there is some subtlety of the English language that could be elucidated by this question, but that is not what has been asked. It's not like he wants to know the difference in usage between "a week after next week," "in the next two weeks," or "two weeks from now."
Only one of OP's three alternatives is textbook English, so basically, as worded, the question boils down to "which of these phrases is correct English?" which is very much ESL and off-topic.
In my book this question is not off-topic.
Fortunately in my mother-tongue the phrasing to the given question is very similar to English so I am able to decide which one I have to choose and know the difference between his examples but I can understand that for a non-native person it can be difficult.
In case of the given question it is very clear to me, but
- My friend and I decided to go to the beach in the next two weeks.
think of this sentence
- My friend and I are on vacation in the next two weeks.
So, I am not sure, if this is possible in English or if there must be for but at least in my native language both are correct (and mean nearly the same).
Independent if in the next two weeks is possible in my example or not, how should a non-native person decide that? And this has nothing to do if the user doesn't seem to have a strong enough grasp of English. If so I have to sign off, too.
For OP's question: I am sure there are lots of non-native confronting this issue or similarly.