I think it would be a very useful skill, to know how to put our animals down in a loving gentle way. They learn to depend on us for everything, and we should know how to put them out of their suffering. Not sure I can get to that point, but I think it is a necessary part of animal husbandry.
Sorry, but there is not a "Loving gentle way" to put down a dog or a cat, aside from taking them to the vet. Period.
There is no "DIY Euthanasia" for most domestic pets, and for good reason. There is nothing that is endorsed by the HSUS, the AVMA, or any professional medical organization.
It is hard enough for the layperson to euthanize a chicken without practice (I am a vet. And my first--and only--cervical dislocation was a terrible, awful, inhumane disaster).
You cannot cervically dislocate, drown, gas, shoot, or any other method that is compassionate and humane. You take your pet to a vet, period. I'm talking dogs, cats, whatever. Chickens are different, if you can stomach it. But that's pretty much it, and in retrospect, I wish I would have taken my chicken to a practicing vet and paid the cash rather than put my beloved hen--or myself--through that again. Money well spent.