Carrying your discs in a shopping bag might work for a while, but at some point it's a good decision to pick either a proper Disc Golf Bag or a Disc Golf Backpack. The following factors should be considered: How many discs do I want to take? How much extra gear do you have? What feels comfortable? Which bag or backpack makes you look badass?
As for baskets some are more or less portable. All of our baskets are sturdy and love to catch discs.