I'm in the same "transporting stuff to and from a kiln that isn't in my house/studio" boat, and have had to do this since 2012.
I use plastic bags from the store, plus those produce bags. I wrap each piece (glazed or green) individually in a bag, lightly, and pack the pieces into an old clay box or a large insulated tote bag (the kind with a flat bottom and a plastic panel in the base to keep it flat). Extra bags get wadded around in the spaces so they don't jostle around in the car. Sometimes I can "nest" one bowl in another, if they aren't too fiddly.
Then again, my work is pretty sturdy...I don't do super-thin wares, because I'm still a beginner and, y'know, beginners do everything thicker than the pros. So maybe that helps me -- I have maybe had one piece break in transit in the past 3 years and maybe one other that broke because I mishandled it while taking it out of the tote bag.
I also only transport about 5-7 pieces, max, at a time. This works for me, because I just go once a week and do a dropoff/pickup over lunch.
As for glazed pieces, I don't use any kind of hairspray or starch or anything other than the plastic bags and the only time I've had chipping is when my glaze layer was unnecessarily thick. I just make sure the pieces have completely dried (usually overnight) before I bag them up.