Yes, you can arrange for the car to both move on a train and to have it arrive at your destination at the same time as you do.
Open the phone directory (yellow pages) and look for Freight Forwarders, or Auto Transport, or Car Transport, as you will find forwarders and brokers that specialize in this service. Finding a specialty company is likely best for you as they will have experience in the move, and can guide you accordingly.
For any choice you make, you need to make the arrangements in advance of your travel date to ensure a timely arrival.
If you choose to send it by train the car will probably travel on a freight train and most long-haul trains are now run as either passenger cars only or freight cars only, so even if you are travelling on a train the car would probably be on a different one, and it would leave at an earlier date, to allow for connections.
There is no set cost as it varies by carrier (different rail companies) and it will vary depending on the actual cities at each end, because not every railroad company will have its own rails and equipment operating between your choice of cities, so connections must be arranged to get your car to its proper destination. All of this will be fully arranged for you, at one cost.
You should contact at least two or three companies to ask for quotes on the service and cost. They will want to know the make and model of your car in order to estimate the size and weight, as this may affect the net cost to you. Get the quote in writing, and also ask for their guarantee of possible damage insurance.
Good luck on your move.
:)