There's no one correct way to cut yourself haha, it's entirely up to preference. Yeah, I'd recommend a new razor for the sake of a cleaner cut doing less damage by being jagged or dull and also for cleanliness so that you don't run as much of a chance of it getting infected. Cut as deep as you feel is necessary for emotional relief. If you haven't done it before I would recommend starting small and light so that you can get some experience with it. But really, be smart about it. Cutting deep can give you some very disasterous results and you'll need to have the proper wound care items (i.e. something to sanitize and something to properly bandage with) to deal with that should you get to that point, or even professional medical attention if its too bad. For me personally I have been cutting for self harm for around 12 years and it's never been a complicated process, even as a kid. You kind of just do it to the extent that you're feeling desperate! These days I tend to not cut super deep but quite numerously, just enough to watch the blood flow down a bit, and I find myself hitting the spots where I cut with a fist in the days after that it's tender and actively healing to extend and control my pain without having to cut too much. Be safe with how you treat yourself doing this, as paradoxical as that sounds. Do your best to take care of your wounds and be certain about your want for this kind of self harm. Scars last for many years and carry a big social stigma.