Fridge
Fresh fish should be stored on ice even in your fridge. Don't let the fish touch the ice. One of the easiest ways to set this up is using a cooling rack that fits into a shallow roasting pan or sealable container. Fill the pan / container with ice, place the cooling rack over the top with fish on top of this, then cover with alfoil or lid.
Yes
If you can't use your fresh fish within a couple of days, freeze it. If you have bought fish that has previously been frozen, most can be refrozen as long as it was defrosted in the refrigerator and has not become warmer than 5 deg. Keep in mind that once defrosted again it will need to be used asap.
Use it up!
Got leftover fish from the night before? The key to eating it again is gentle heating. Use it up in fish tacos, fish cakes or flake it and stir through the final stages of cooking a chowder or soup.