I used Horseless Carriage Carriers
http://www.horselesscarriage.com/ to transport my Elise from WI to GA and later Minuteman Transport
http://minutemantransport.com/ to do the same with my Elan. Both of these transport companies used fully enclosed trailers with flat loading lift tailgates and the cars were held in place with nylon straps on the wheels (no chains) so they arrived with no damage and in the exact same condition in which they were loaded. Both companies delivered the cars with door to door service and were very careful with the way they handled the cars. I had no problems with either transport company other than it took longer than expected for Horseless to pick up my car due to the lack of available space in their transporters because of the large numbers of cars that they were transporting at the time. The Elise shipped for about $950 and the Elan about $650 so you have an idea of what it should cost you.
No matter which transport company you choose, the main things to check are if they will remove your car from the trailer at any time to make room for other cars during shipment and if your car will stay on the same transporter with the same driver for the entire distance to your destination. Always try to hire a transporter that will load your car, tie it down and not handle it again for the duration of the trip which also ensures the same driver/transporter for the entire trip. The less handling your car receives during transport the less chance of damage to it. Also, if the trailer holds the cars on upper and lower decks make sure to have your car top loaded so there is no risk of a car riding above yours leaking fluids or dropping loose parts onto your car. The better transporters will always do a meticulous visual check of your car at the time of pick up for any preexisting damage to your car with some like Horseless Carriage noting even the slightest blemish (they even noted my Elise had a tiny smudge of excess wax under the front end which they recorded as paint damage!) and have you cosign the form with the driver at the time of pick up and then do the same once the car is delivered and reinspected. This ensures they deliver your car in the exact same condition in which it was picked up. Most of the transport companies these days also offer tracking so you know where your car is at all times which offers some peace of mind as well.
Dan