If you have a lot of stuff to move, you will have to ship it. The amount of stuff you have will determine the type of shipping you do.
To find shipping companies and personal experiences (from a UK point of veiw) have a look at this thread and this wiki page: UK Shipping Info
Types of Shipping
Freight shipping
is best when you are going to move a whole house worth of stuff, furniture and all. Most often you will be using a full container.
Excess Baggage
is the service you need if you have a reasonable amount of bulk or heavy items, too much for regular shipping services, but not enough for a whole container. This kind of shipping can be done by air or ship. Air freighting is usually charged by weight, ship freighting is usually priced per volumetric weight.
Air Cargo
You may sometimes find that simply paying for extra baggage with the airline your flying with is the best option. You will have your stuff immediately!
Customs
Generally US citizens do not pay any import fees if they are importing their own personal items, not for commercial purposes. To avoid paying import taxes you will have to provide a copy of your US passport, or immigrant visa to your shipping company.
In order for your goods to clear customs you will need to fill out form 3299. You should provide this to your shipping company if they are going to clear customs for you. Otherwise you will fill this out at the customs office to clear the goods yourself.
Here is a good guide to US customs by Anglo Pacific.
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NOTE: The above information does not address the specific requirements for any given case and is not a substitute for the advice of an attorney.