Having the auto transport driver sign the bill of landing is definitely the first step toward toward securing your argument that your vehicle was damaged during car delivery. It is always wise idea to take pictures of your vehicle before auto shipping and then again when your automobile gets delivered to its final destination.

I agree with john, in that most auto shipping companies will want to handle this before it escalates to a bigger matter. Also if the safety of your vehicle is a concern you may want to spend alittle more and get enclosed car shipping service http://www.karshipping.com/car-shipping-blog/category/enclosed-car-shipping/

The estimated car shipping rates that most companies are around $150 - $200 deposit depending if you are getting open carrier or enclosed car transport.  Keep in mind that an enclosed car transport may be more expensive. I got an auto transport service from a transporter company and they did not charge anything upfront. 

When a car shipping company does charge a deposit upfront, they normally require that you pay the remainder of the cost when the auto transport carriers delivers your vehicle to your location. If you’re not charged by the auto transporters, then the whole amount will be do at time of delivery.

Another factor is definitely the distance your car will have to travel. Shipping car from New Jersey to Alabama is faster then transporting a car from New Jersey to Arizona.   Many auto transporters do offer express delivery.   Express car delivery does cost more money.