There are several reasons one might wish to relocate a car across the country. A new purchase, a sell, a job relocation, an extended summer vacation, Snowbirds, are just a few. There are several considerations to think about, such as driving the car, selling the car, hiring a driver, paying a carrier, and etc. The cost to do each should be carefully considered including wear and tear, fuel costs, hotel costs, toll fees, unexpected mechanical failures and a bunch not listed here.
As far as I know there is no way to move a car across the country without it costing you in some way. Even if you had a friend that wanted to go across country and is willing to drive it for free, pay for the fuel and pay his own expenses, there is still the wear and tear on the car, the depreciation due to the overall mileage increase, not to mention use the mechanical repairs that could be encountered. At this point it would have to be a very good friend, or someone desperate to cross the country without flying or taking a bus. Either of these would be a cheaper for him or her than driving you car at their expense including repairs.
There are plenty of articles on the internet about the preparations one needs to have your car moved either by rail or truck. The criteria for selecting a carrier, and what to watch out for while making that selection, so I am not going to go in to all those. Likewise there a multitude of options to decide such as open closed, multiple carriers, single flatbed trucks, terminal to terminal, or door to door.
Inclusion, there is a lot of decisions and options when deciding to move a car across the country. The longer distance, the more expensive and the more that can go wrong.