I've tried to be as honest as possible writing this, and it's been hard to write, so I'm sorry if anything I have written is upsetting.
You should be allowed time to sit together in the room first and say goodbye. You will have to sign consent and they will ask if you want to take him away afterwards, or for cremation. They'll then shave a patch of the leg - my dog got really distressed at this point and in retrospect I wish I had thought to take in some sort of "treat" - even something that's usually forbidden but really tempting like chocolate buttons. At the time all I could think was "he knows and he doesn't want it.." but actually, years down the line, he was fighting something scary at a time when he felt vulnerable.
They put him up on the table and put the needle in - and it was pretty instantaneous. He went limp and we laid him on his side. The vet told us that sometimes they can growl or empty their bladder, but that it's reflexes.
When it was over, the vet left us alone with him and allowed us all the time we needed. We took his collar and lead off, gave him loads of kisses and got ourselves ready to leave. That was the hardest bit, actually - just leaving him there. I wish that I hadn't looked back as I was leaving, as he looked so alone.
I'm really sorry you are going through this. My old boy died 7 years ago now, and I'm still welling up. It hits you in the most strange ways - for years afterwards I would have the same dream where the injection didn't work, and the vet would tell us to take him home instead as he was skipping around. I would wake in tears.
I was given an option with my cat to just be with him whilst they knocked him out then they euthanased when he was asleep and i didn't stay for that. That may be an option for you. It was just too hard for me.