I use a rubbermaid tote to carry my machine and fabric. I have a nice fishing lure box I use for my notions and I generally keep it well stocked since I take it to other sewing groups. I carry all this using a luggage carrier. The rubbermaid tote goes on bottom and the fishing box on top, then the entire thing wheels around easily. The advantage of a plastic tote over a soft carrier is that the entire thing is completely waterproof, something handy in my climate!
I do always check the supply list for the class, but I usually add a few things based on experience. I always bring good scissors, thread nippers, seam ripper, marking supplies, lots of bobbins, threads, and all my sewing machine feet. If it's a quilt class I bring rotary cutting supplies, even when the list says the class includes them. There is always competition for the rotary cutters and mats!
Pam