As long as the tank is secure and protected on all sides that might come into contact with something, it will be fine with careful driving. styrofoam is a good choice as a base, and try to keep that as even as is possible. Anything soft, like cushions to pack down the sides, back and front are fine.
As for the fish, bagged with plenty of room for oxygen at the top. Put them in a cool box, which will insulate the water and keep it warm. Don't blow into the bags though, this is just filling the space with carbon dioxide. Better to just tie them without anything, but if you can get oxygen then that's best. You can ask your LFS for fish bags. They usually give them, but might charge a nominal fee.
Plants will be absolutely fine wrapped in damp kitchen roll, or similar water absorbent material. Not toilet paper though. It just disintegrates and you'll have a job picking off little bits when you unwrap. Be careful pulling them up you don't damage the roots too much.
Don't forget to check your water parameters in the new location, before filling up and adding fish. There might be a sizable pH swing. Take as much of the tank water with you as possible and get your fish back in asap with that. Buy some containers if necessary for that purpose. I think it's the best way to relax your fish after the trauma. Do it as you would with new fish. Get the tank up to temp and place the bags in the tank to acclimatise before releasing. Age your new water if necessary, and if there's a pH difference of more than .3, then you'd be better off doing smaller water changes, like 20%, 30%, 40% over a period of days, increasing the amount each day to get them used to the higher/lower pH.
Good luck with it. Rob