On my boat, I apply the wood shampoo immediately (so they don't flop around and bruise themselves), then bleed them (in my multi-purpose square bucket, to avoid attracting sea lions). In warmer weather, I will then gut them. Then they go into an icewater slurry, either in the fish box or into a cooler.
When I get home, I fillet them, remove the rib bones, and usually steak my fillets. Since they lay flat in the fish box or cooler, the stiffness helps me fillet them.