Avoid storing the bags in direct sunlight since the bags could create humidity that damages the pieces.
You can buy mesh storage bags online or from any big box store.
You can usually get bins that interlock and stack so you can keep them even more organized. You can keep multiple puzzles in the same bin if you have them separated into different storage bags.
If the pencil cases have metal grommets, secure them in a 3-ring binder. That way, you store all of your puzzles in one place and can easily look through them.
You can buy photo boxes from your local big-box store or online.
Opaque folders make it hard to see what puzzle is inside, so choose ones that are transparent instead.
You can get stacking baskets or containers online or from your local big box store.
You can buy a wire puzzle rack online. Otherwise, a wire document rack from an office supply store may work in a pinch.
You can get puzzle mats in a ton of different sizes, so you’re bound to find one large enough for your puzzle.
If you don’t want to cut the box cover, take a picture of the box and print it out. Some puzzles come with a small sheet of paper with a picture of the completed puzzle.