Visit the web pages of several major trans-Atlantic airlines. Most major companies suggest a preferred baggage-courier service, and may even offer discounts for booking through the airline. Visiting a few different sites will allow you to comparison shop.
Purchase a light but spacious suitcase. A heavy leather suitcase may add to the cost of sending luggage overseas. Also, a lightweight suitcase is an investment that may pay dividends for future travels.
Purchase vacuum storage bags. Placing items in vacuum bags can free up an enormous amount of space, enabling you to fit far more into the suitcase and possibly eliminating the need for extra suitcases.
Check the weight allowance for each bag before packing. Any excess weight will either incur a higher fee or the courier service could even refuse to fly the item. In the latter situation, it is unlikely that you will receive a full refund as time, energy and money will already have been expended by the company.
Pack only the essentials before using the courier service; remove any heavy items that do not need to be sent home. The lighter the bags the cheaper the service will be as the rate is usually charged by the kilogram as well as for each suitcase.