A week passed by so fast. I remember just asking my nephew "what's today" and he said "It's Friday". Now, it's Friday again. OMG, a week passed already? I don't even remember what I did during the past 7 days. I would get rid of the second "and" because the sentence becomes too long. Also saying "what day is today?" sounds a bit more correct. In spoken English people say "What's today" so its not that big of deal

Nice work. I would write it this way The week passed by so fast. I remember just asking my nephew "What's today?" and he said "It's Friday." Now, it's Friday again. OMG, a week passed already? I don't even remember what I did during the past seven days.

You can put a comma before quotation marks. Also: "a week has passed by already?" But those are just minor things. Well written. Cheers.