I think it is a nuisance to pack for classes. I have had to buy a rolling case for my machine, with exterior pockets for stuff. (Joanne's with a coupon) I have a regular class on Monday's so now I know what they've got (irons) and what they don't (a decent cutting mat - theirs is chewed up!)
What I do is sit at my table and grab everything I would normally be using for whatever tasks I will have to do at class. I pack extra needles, extra thread.
If I have fabric I have already cut out, I bring extra pieces as I find it hard to concentrate in a room full of happy chatty people who all can quilt in their sleep and use their machines blindfolded, and I sometimes get distracted and miscut or mess up a piece of fabric.
I fold a quilt over a padded coat hanger and cover it in a drycleaning bag.
Small tools and stuff, and cut out pieces of fabric, I put in cheapie food storage containers, and I put folded fabric in large plastic bags unless there is too much and then I use the coat hanger.
Once you get the hang of it, it gets easier.
My machine repair person told me not to leave the machine in the car overnight regularly if it gets down to freezing. I had thought maybe I'd bring my old clunker to classes and leave it in the car, and keep my lovely little Janome safe at home, but no such thing.
Good luck - it is a lot of fun despite the packing!