I started a PhD start of October - and it quickly became apparent that the department I'm working with is very chaotic. Now, almost 5 months after I have no project, no research plan or whenever I try to talk to my supervisor she says there's plenty of time and we'll see what happens, because she is applying for some funding at the minute. A post doc quit recently so now, I am mostly doing and continuing his work, which is fun but not a 3-year project. My supervisor is quite nice but very dismissive and laid back. She oversees 4 PhDs and 2 Postdocs as well as some masters and undergraduate students which I thought was a lot?
I have applied for another PhD, starting in October next year as I don't want to be left with no options if in couple of months it all goes bad. Now, I got accepted for the new PhD and it is a very good project and I might want to take it.
My issue is I don't know when to tell her? I hope she will be understanding, but if she's not can she kick me out before the end of this academic year? I don't want to tell her a month before I go but I don't want to be stuck for 6 months as I need to live off something before the other one starts.

Thank you!

How to quit? Just tell your current supervisor that this PhD is not for you and that you quit. Apologize and thank her for her time and wish her the best for the future. There are more students than PhD positions so she should be able to get another very quickly.

You do not need to tell her where you are going. Some grudgeful profs contact the new employer or PhD supervisor to badmouth their former student. You don't want that. Quiting is usually pretty immediate after you inform verbally and then send an email to your uni. Maybe you need to tidy up your desk and lab feezer, but all in all, you should leave the place in a few days tops. Meanwhile, you can get a temp job till you start your new PhD.

Agreed. I would hand my formal notice in first thing tomorrow morning.