I'd be lost without my little yellow wagon (the flat-bed/you can have sides or not garden wagon version), and my hand-truck. If I'm just shifting a couple bags of dirt/mulch/compost, the hand-truck is easier. But the wagon can move up to 6 bags at a time.
For weeding, I either use 10gal rubbermaid totes, or a 45gal wheeled trash bin to put the weeds in as I go along.