Get some of those storage containers that stack and have lids. Put Newspaper in the bottom as I would think they would b the heaviest item. Or better yet get those containers that are bins and lock together. I think they hold potatoes, onions and the like in your kitchen. They have an opening at the front for easy removal and placement. If I recall correctly they have holes through out for air flow so I would put some strong bags inside that don’t break easily. Or another container.
You might want to get a small garbage can for each recyclable category, (A larger one if you have a greater quantity of one type of recyclable item.) Store them side by side with bags inside to hold the items. This keeps the cans clean and you can easily remove the items then. Don’t forget to clean out the items you are recycling. Otherwise you will have an odor and possible pest problems. I used a garbage can to hold aluminum cans in until I was ready to take them to the recycling center. I kept it on my back porch and covered it with a pretty old sheet. (You could make a cupboard around them to hide them or something else.)