Is this grammatically correct? "집에 가고싶지 않아."
is this correct? "집에 가고싶지 않아." or do i need "난" in the front?

Question

Answer

It's correct. You can put 난 but it's not necessary. 
Don't use 난 too much, just once in a first sentence of paragraph.

나는 오늘 학교에 갔다. 학교에 가서 열심히 공부했다. 
학교가 끝난 후 집에 돌아왔다. 배가 고파서 밥을 먹었다. 

If you use 난 or 나는 in front of every sentence, it becomes childish. 
hmm It's better not to use 난 in this paragraph..just minimize it!