I have shipped and received barrels in a few different ways. If it’s enough barrels that you have a dedicated trailer, you can load them in standing up on their head loose in the trailer. Use a load lock or a strap to keep them from shifting front to back. I’ve also received them on pallets standing up. Usually they are 4 to a pallet that is 48 inches by 48 inches, and some strapping is used around the 4 barrels to keep them from shifting. I’ve also seen barrels shipped on racks with the barrels strapped to the rack, but racks are expensive.
You’ll want to make sure your withdrawal bond coverage is high enough.
I’ve never used plastic wrap, its not necessary but it won’t hurt.
As far as Hazmat shipping goes barrels are UN3065 FLAMMABLE III packing group 3 as long as the liquid is below 70% ABV. If the liquid is 70% ABV or above, then the hazard class becomes FLAMMABLE III packing group 2.
We had a barrel get lost on its way to the distillery once, and that was scary, but the carrier found it and delivered it about a week later. I don’t think any extra insurance is needed, but when you line up the freight you can list the value and make sure you’re covered.