Basically you will make a simple tote rounded on the bottom and big enough to hold the bike. I would make two u shaped pieces big enough for the folded bike to go in. Make it out of heavy fabric, maybe canvas. Although a quilted bag would be heavy enough. If you want a quilted one, then make four u shaped pieces and add batting, quilt both sides, then sew them together. Use that webbing strap for the straps, unless you want to quilt straps. If you can get to a sporting goods store that sells rock climbing stuff they have a great strong strap material. Make two straps that goes from the bottom of the bag, makes a u and goes back to the bottom of the bag( one for each side of the bag). Sew the strap up the side to the top of the bag, like this:
http://www.mayarts.com/blog/diy-tote-bag/ The straps will be stronger and help support the bike. Hem the top or add binding. You could do a tap with a snap or a button to close.