I am in the middle of shipping a Model S from PA to CA, and I've learned a lot about shipping in the process.
Almost all shippers are brokers. They work as a liasion between the carrier (who owns the trucks and usually doesn't deal directly with the public) and the customer. If you do a search for auto transport, 99% of the companies that turn up will be brokers. Most brokers will post your car on a computerized system called Central Dispatch, which is watched by a majority of trucking companies. Your car listing will show the origination and destination cities, the distance between the two, the total miles and the price the broker is offering per mile. It's very easy for the trucking companies to see who is offering the most $$$ per mile, and those are the cars that get picked up first. Apparently, there are more cars needing to be moved than there are trucks to carry them, so it seems to be a truckers' market.
What this means for you is the lowest priced quote from a broker isn't necessarily the best deal, because the broker cannot guarantee your car will be selected by a carrier. All they can do is post your listing on Central Dispatch and hope it attracts a carrier at the price it's being offered at. In my case, I was looking for enclosed transport from PA to CA. The broker I chose quoted me a price of $1895, and then listed the car for $1300 (50 cents/mile), hoping to keep the $695 difference as the brokerage fee (which, by the way, is very high - the standard brokerage fee is around $200). My car was one of 18 cars in PA needing enclosed transport to CA, and it was ranked 16/18, right near the bottom - no one was going to move it when they had 15 other cars going in the same direction that would pay them more.
Long story short, my broker cut his commission and raised the listing price to $1700 (putting it in the top 25% of listings) which attracted a carrier within a day. So the cheapest quote may not be the best choice.
Also, beware of brokers. They are largely unregulated, and the industry as a whole doesn't have a great reputation. There are a lot of horror stories out there. Don't pay anything to a broker in advance - wait until they've found a carrier before you pay their fee. Ask the broker to supply proof of insurance from the carrier. If your car is listed at too low a price and doesn't move, the broker will either have to raise the price he quoted you or lower his own commission. Or you can just wait. And wait, and wait. And hope someone comes along with a truck who's willing to move your car for the low ball price. You get what you pay for.
Finally, I'd recommend using enclosed transport rather than an open car carrier, even though it's more expensive. It not only protects your car during the trip, but the enclosed carriers are used to transporting expensive cars and seem to have a higher level of care and service. This is my first Tesla, and even though it's 5 years old, it's the most money I've ever spent on a car and I don't want anything to happen to it on the way to my house!
If you're wondering about reputable transport companies, I've done a lot of searching across multiple forums and the ones that get mentioned a lot are PlyCar, Reliable, Interstate, Horseless Carriage and Gran Turismo Motor Sports. All are expensive. To this list, I would add Capitol Auto Transport, because they were so honest and transparent, and really went the extra mile to answer my questions. (I had already committed to a different broker before talking to them, so they were not who I used.)
Hope this helps!