My suggestion would be to get really drunk because then your fear or anxieties turn into carelessness and it's easier to make the cut. This is just in my experience.
To bleed out you would need to cut an artery, probably the radial or ulnar if you're going to wrist/forearm. Check where your pulse is to find it and check online to see pictures of the anatomy so you know where to cut. You will be more successful if you cut vertically.
For depth, it kind of depends on the razor and your state of mind. Get the sharpest one you can, I use the ones you can buy in a pack of ten that are used for exacto knives. For me, I do the cut quickly instead of a slow one. The artery isn't too far down but you need to get through epidermis, fat, nerves etc, so push down as much as you can. If you need to you may have to go back in the original cut to cut down more which I know if really scary.
I wish you peace & comfort in what you decide to do.