As Mike has pointed out, not all delivery captains will allow you to come along, especially as they need to have crew that will allow them to get the delivery done as quickly as they can, and you may not have the skills or experience required by them.
Getting your sailboat, and putting it on a ship to bring it over to Europe is going to be fairly expensive. You should probably contact Dockwise
to get a rough idea of the costs involved.
Also, be aware that you will need to pay for the dock fees in the USA until you can get your boat onto the transport ship or the delivery captain takes the boat out to sea, and in many states, if you exceed a certain time period, you'll also be responsible for paying taxes on the boat in the state. You should keep this in mind when looking to purchase, since certain times of the year are better than others for crossing the Atlantic.
A delivery captain is probably less expensive than shipping the boat across, but comes with more risks to your boat, since it will have to make the journey across the ocean on its own power, and may be damaged in the process.
Of course, a lot of this depends on what boat you decide to buy. If you don't get a true bluewater passage maker, then shipping the boat, rather than having it delivered would probably make a lot more sense.
Omatako, one of the posters on this forum, recently bought a boat in California and sailed it back to New Zealand, IIRC. This would be your third option... sailing it back yourself.