I like to make a sample block first, then I cut out all of my pieces. If it is a two block quilt, I may just cut one blocks pieces all at once and then sew them all before going onto the second block. Cutting is not my favorite part, so I tend do get 'er done and move on to the fun part :D
I will store the pieces according to how I will be chain piecing them. This way I can pick up the container with just those pieces and sit down and sew, grab the next set and repeat. I may store them on paper plates, pizza boxes, plastic containers, whatever is handy. I also put a piece of paper labeling what each piece is, in with the pieces, so I don't get them confused, or ME LOL