What is the most logical next event?

How to help a friend grieving the death of a dog
Offer to go to the vet with them.
If the dog is going to be euthanized, you can offer to go to the vet with your friend. You can hold their hand as their dog is put to sleep, or you can wait in the lobby.
Having someone there for them can help ease their pain. You can say, " do you want someone there when you take fluffy to the vet? I'd be willing to come with you, if you want.