“After he comes back, we will begin to cook" is colloquial/informal, whereas "After he has come back, we will begin to cook" is more formal and better suited for writing.

Hi Yu: In the first sentence, you have correctly used the simple present verb tense (comes). In the second sentence, you have used the present perfect verb tense, which is generally used to describe actions that are complete with respect to the present time (e.g., "He has eaten breakfast.").