I’m so sorry.
At home is better for your pet, but harder for you, as later you have to often see and maybe even walk through the space where they died. So think carefully about where in your house it would be. I foolishly chose the playroom and it’s still hard to be in there sometimes.
The usual process is the vet gives your pet a quick injection basically anywhere, which is a sedative. Ours was jabbed into the back. Then the pet falls to sleep and the vet shaves a bit of fur off one leg and inserts a canula (a hollow tube) into a vein. Then they ask if you’re ready for the final injection. When you tell them that you’re ready, they inject into the canula a drug that stops the heart: it works very fast and they tell you the pet is dead a second or two later.
(Some vets don’t bother with the sedative first, so check with the vet that they will do it.)
I couldn’t talk about it without crying so I made most of the arrangements via email which the vet was ok with.
It was so much harder than I expected and not as peaceful as I’d been led to expect (angry cat) and for a couple of weeks afterwards I had genuine PTSD style flashbacks. So be very gentle with yourself through this process. The loss of a pet was harder on me than the loss of some family members. It is such a responsibility to carry. But we do it for them. Keep reminding yourself that you are doing the best possible thing for your friend and making it all as peaceful and painless as possible and that if there was a better option you would do it. Sometimes none of the options is good.
It might also help to think about your own beliefs about death. I’m not religious but with both of my pets I was surprised by the very strong sense that they lingered near their familiar places/people for three days after death before drifting away. You might have some vivid dreams about them, soon after they die, and they’ll probably be young and healthy in the dream. That’s normal and what you think it all means is up to you. I also felt intensely that it was important the body be undisturbed as much as possible. My vet kept stroking my dying pet and although she meant well that upset me a lot as he hated strangers touching him. So maybe let the vet know any particular requests you may have, I wish I’d asked them to touch him as little as possible.