A bike is a great idea! I also agree that the list will vary by scenario. For a major collapse/true SHTF where we'd burn our cash & credit reserves, I think my priorities would be:
* cooking supplies (tools like pots, pans, knives, spoons, etc. that might be hard to replace or repair if supplies disappear)
* gardening supplies (again, tools like hoes, shovels, spades, hoses, etc.)
* food (specifically long-storing and/or aids to cooking like spices, flour, evaporated milk, etc. - note: I have food supplies stored, but every extra bit helps)
* shoes and clothes
* sewing stuff (fabric, needles, thread, zippers, etc.)
* animal stuff (feed, bedding, medicine, etc.)
* fuel (gas cans, gas, diesel, wood, etc.)
* Hardware (hinges, hooks, screws, nails, paint, etc.)
* Camping stuff (tarps, sleeping bags, cast iron, fishing hooks, etc.)
* misc (books, board games, plain paper & pens, anything not nailed down that looks useful)
Basically I'd hit - in order - grocery store, Home Depot/Lowes, Tractor Supply/Petsmart, Dicks/Bass Pro, then anything I can think of that's not mobbed.