Learn English – “In the next two weeks” vs. “next two weeks”

Which is correct depends on what is meant.

"They'll quite happily squander a whole year's savings on two weeks in the sun" means they will squander the year's savings by spending it on a holiday; a package deal, perhaps, or an expensive hotel.

"They'll quite happily squander a whole year's savings in two weeks in the sun" means that they will squander the year's savings during the two-week period; on gambling, perhaps, or some other pleasure, and for some reason the sunniness is relevant (heat affecting the brain, perhaps?).

It depends what you mean.

"I'd like to schedule a meeting with you next week" could mean that sometime next week, you'd like to schedule a meeting. That meeting might actually be scheduled for next month, but next week you'll agree to the time.

"I'd like to schedule a meeting with you for next week" means that the meeting should take place next week.

Edit: "I'd like to schedule a meeting with you for next week" could also mean that you want to schedule a meeting to talk about something that will happen next week. The actual meeting will happen this week, and you want to talk about next week's event.

That's just one literal interpretation, and the context of where and when you're saying it might make it more obvious that you mean something else.

Edit: You've just changed your question, so I'll add this:

"I'd like to schedule a meeting with you in the next week" means that sometime in the next week (which includes whatever is left of this week), you'd like to schedule a meeting. That meeting might actually take place next month.