Some people mentioned the carotid artery as the best bet, but I disagree. Typically the bigger the artery is the deeper and more difficult it is to cut, but the faster you bleed out. If I were you I wouldn't fuck around with a knife in my throat where I can't see what I'm doing, I'd just go for the Radial artery in the wrist or the Bracial artery on the inside of the elbow.
Radial:
"Walk up the road, don't cross the street."
Cut from the wrist under your thumb (where you can feel the pulse) up towards the elbow, NOT across to the other side of the wrist. You'll have to go through the skin (white), fat (yellow, looks like baked beans), the Palmar Carpal ligament (red or white I guess) before you reach the artery, wich has muscle around it you also have to cut through.
Bracial:
The Radial and Ulnar arteries before they split off, so it's much larger.
Cut across your arm on the inside of your elbow, in or just above where your arm folds, to the right (on your left arm) of the tendon connecting your bicep to your Radius/forearm bone. Again, feel for the pulse. You'll have to go through all the layers again exept the ligament, wich could make this easier.
This method is EXTREMELY painfull, as others have mentioned, and you need time to bleed out. I've tried for both arteries multiple times and failed miserably. My best advice is to just go for it, slow and steady DOES NOT win the race, or just don't do it at all.
If you decide to go through with it, I wish you the best of luck. You'll need it.