I interviewed for a job in the start of the year and didn't mention that I had applied to graduate school the previous fall and would attend school if accepted. I got the job and started this March, but 3 weeks in, I was notified that I was accepted into my dream school. I plan on leaving after 5-6 months-when should I bring up my resignation and should I mention the specifics as to why I'm leaving or is it alright to just say it's for "personal reasons"?
Resign when you are at your notice period as per your contract. It doesn't matter that much what reason you give.
Personally I would just tell the truth that I was going back to school. It's a perfectly good reason to do so, and most employers are supportive of the idea of people furthering their education. But so long as you're not breaching your contract you can leave with whatever reason you want.
You're just starting in the workforce, you won't be difficult to replace.