I am flying to Spain tomorrow and am staying/will stay there one week.
What is the best solution I think both are valid.
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In your example, since the occurrence is soon
I am flying to Spain tomorrow and staying for a week.
I am flying to Spain tomorrow and will stay in Madrid for a week.
otherwise, you might use
I am flying to Spain for Christmas and will be staying for a week.
I am flying to Spain for Christmas and am staying with my brother.
Using either of your choices would be understood.