I've tried many things over the years for taking everything needed to class. The best thing I've found is the Gotta Have It Bag at allbrands.com. There are two pieces to it, it's on wheels, and it has many pockets with zippers. It holds my machine, but you could use that big space to hold your quilt top in progress. I also use a multi-drawer fisherman's case for notions storage and always have whatever I might need. Check at Lowe's and Home Depot in the tool area for other containers with drawers and on wheels.
Get a vinyl tablecloth with flannel backing and some skirt hangers, the kind with two metal clips. You might need several hangers. Using the flannel side, put your blocks on it for a design wall and hang it up somewhere. You can roll it up when you need to and the pieces will stay put.