I've used two companies I would recommend.
Reliable Transport has moved two cars for me. One from Atlanta and one from Alabama, both to Los Angeles. Both cars arrived ahead of schedule and were delivered by husband / wife teams who were car nuts. Great attention was paid to the safety and well-being of the cars during transport and unloading.
The other outfit was TFX International. They were very friendly and the driver was a great guy. The car did arrive a day late but it wasn't their fault. Seems the delivery before me said there was room to turn the truck around, but when the driver got there it was a very narow road. So he had to back the truck out..... 20 miles!!! And it was a twisty mountain road, too.
Another company I've heard spoken highly of is Horseless Carriage. They specialize in collector and exotic cars. I have no personal experience with them though.
Just be sure you don't use one of the brokers. Call the transport company directly for the best service and price. Also, go with an enclosed trailer if you can, unless the car is a bit rough and you don't care about rain, hail, road debris, etc.