Is this sentence correct?
방학동안 제가 서울에 가족 보러 갈 거예요
“During the holiday I will go to Seoul to see my family”
Is that right?

Question

Is this sentence correct?
방학동안 제가 서울에 가족 보러 갈 거예요 
“During the holiday I will go to Seoul to see my family” 
Is that right?

Answer

Yes, correct! Or you can say '저는' instead of saying '제가'. 
'저는' sounds more natural unless you're asked like this. 
"방학 동안 누가 서울에 갈 거예요?"

방학 동안 저는 서울에 가족 보러 갈 거예요. sounds more natural for saying your plan during your holiday.