I have a box I save all my scraps, strips, etc. in. I separate the childrens fabrics from the rest of the scraps.
When I start a new scrap quilt, I look here first to see what I can use. I don't keep anything under 2 inch squares.
I keep them in stacks so the fabric is neat and not all wrinkled. I never cut my scraps into pieces because you never know what you will need. But if I have leftover pieces the same size, 2 X 2 ex., I store them together in a ziplock bag.