I don't eat salmon, but I have filleted many fish. Mostly grouper, hogfish and various snapper. I start by cutting down to the backbone from top to bottom, behind the head and pectoral fin (same on both sides). I then run the knife against the bones above the spine on the dorsal side from head to tail. I do the same thing on the under side starting at the vent to the tail. Then the knife is run on top of the spine (backbone) to the tail. This usually splits the tail fin, giving me a handle to run the knife toward the head along the spine, cutting off the rib bones in the process. Repeat on the other side. The split tail fin gives me a handle for skinning. I never leave the skin on, don't like the texture or taste.
I don't screw with pliers to pull ribs and pin bones. I just cut off that whole section and not worry about any bones I might have missed. I use the rib sections and carcus for stock.
I check the half salmon for bones and pull them out with pliers. If I buy a whole salmon, I do not fillet it, I cook it whole.
We buy from the fish shop and it is already done for us
My family's method for cooking a salmon to eat cold is as follows :-Hey,
How do you cook your whole fish??
I'm a big fan of steaming inside an aluminium parcel. I score the skin, rub the fish down with EVOO, season her up, slices of lemon and a few bay inside the belly cavity. Maybe lay the fish out on fennel if its available. Best thing about it is the fish cooking in its own juices, no loss of flavor.