I am going somewhere for the summer.
'for the summer' indicates a period time of the summer through which your staying somewhere extends. It maybe the whole period or only just a certain part of it. You are going somewhere for your summer vacations.
I am going somewhere in/during the summer.
Here, the difference is that with 'in', no duration is suggested. On the other hand, 'during' suggests duration, but it does so unspecifically.
Basically, the three sentences mean the same thing:
You are going somewhere for your summer vacations.