To make a synopsis of what's already here, and assuming you have decided your preferred method is either with a handgun or a knife:
1) Using a handgun aimed at your head has about 97% success rate. "At your head" includes mouth, temple, behind ear --your entire head. I suspect the only reason this figure is not 100% is because "to the head" includes aiming at the temple, which has a (relatively) high failure rate. Aiming into the mouth has a very high success rate.
2) A 9mm is plenty powerful enough, especially if you're using hollowpoint ammunition.
3) Exsanguination --cutting your wrists, neck veins, even femoral or brachial arteries, has a very low chance of success and would be very painful.
4) Likewise stabbing yourself in your heart isn't very effective --it has a fantastic dramatic value if you want to make a statement to your unfaithful spouse, but it otherwise has little to recommend it.
5) When using a handgun, be careful about your aim, in your mouth. Have a look at the diagrams in the gun megathread, and then practice your aim in the mirror. In my opinion, you would do better to aim too low than too high: if you miss the brainstem (your priority target) you would do better to hit the spinal cord beneath it than the less-important brain above it.
Conclusion: a 9mm handgun aimed into your mouth is a very reliable method and the best of the options you have chosen to explore.