Coolers are the preferred method for most. Some people use the rolling tote coolers that you see at ballgames...others use handheld coolers or even Styrofoam shippers.
Personally...I just use an insulated backpack with a second insulated drawstring "track bag" inside.
Nope...nothing inside. If it's cold out, it's best to bring the cooler inside with you so that the temp of the corals doesn't drop when you place them inside.