Fish are not meat. Remember that and you will be on the right path toward taking good care of the fresh fish you bring home. And take care you must, because fish are both expensive and are among the most perishable foods we eat.
Doing it right is not difficult, but it is best done with a little extra equipment.
First, you need a large plastic container. Ideally you also will have a slightly smaller container that fits entirely inside the larger one. If you take this step, drill or punch many holes in the bottom of it. This smaller container will hold your ice and your fish. If the smaller container will touch the bottom of the larger one, you will need something to keep it elevated.
What's the point of all this? You will need to keep your fish on ice -- even in the refrigerator. And ideally, you will keep them with their back fins facing up, as if they were still swimming. The reason for this is because unlike meat, fish will rot quickly, even in the fridge, unless iced.
Why would this be? Because fish generally swim in water that is colder than air, sometimes a lot colder. They have evolved to survive in water that can be just above freezing. Cod are the most popular fish like this.
This is why fish fat is so good for you. Because it is liquid even at very low temperatures, it does not gunk up your system the way a highly saturated, dense fat such as beef suet does. Fish fat is also loaded with omega-3 fatty acids, something most other meats -- with the exception of grass-fed animals -- lack.
So here's what you do when you bring your fish home. Make sure the smaller container is up off the larger one, then fill the smaller container with crushed ice. It must be crushed ice. Large ice cubes will take too long to melt and can discolor the skin of the fish.
Put the fish in, bellies down, backs up, cover the container and you're done.
How long can you keep a fish this way? It depends. If you caught the fish yourself, you will get as much as a week to 10 days. A market-bought fish won't last that long, nor will a piece of fish such as a fillet or a steak. Those will remain at peak condition only a few days.
Replace the ice as it melts, and pour off the melt water.
One more thing: If the fish is lean, such as bass or cod or walleye, you get an extra day. If it is oily such as salmon, trout or sturgeon, take a day off the total time. And of it is a really oily fish such as a herring, sardine or bluefish, eat it now -- it won't get any better tomorrow...