The most important thing to worry about is the Bill of Lading or "BOL". It is the document used to depict the condition of the vehicle. It is the only document that can be used to prove damage was cause during transport.
Even if you take video or photos, we cannot force a claim without that form. Bottom line, if it's not there, the insurance says it doesn't exist. The BOL is a bumper to bumper inspection of the vehicle's condition at bot the pick up location and deliver location. You will receive a copy either by snapping a picture with your phone, a carbon copy if using a regular paper form, or an email if the driver using a tablet.
For more information, you can visit our detailed article about the bill of lading.
At delivery look carefully. Two sets of eyes are better than one. Remember, the carrier may not mean to miss something, but we've noticed the carrier will be very detailed at the pickup where they are most liable, and assume that all went perfectly and do a quick inspection at delivery only because they want to finish quickly to make the most of the hours they are constrained to working.
It may be inconvenient especially if you and the carrier are in a hurry, but be absolutely sure to do the walk around the car with the carrier noting damages. Once they leave, you cannot make a claim.