It depends on what you mean by "ship baggage." A suitcase will be expensive to ship unless it is soft and can be crushed down to next to nothing. Shipping prices depend on both weight and size.
The post office will have the best rates. It takes the longest so don't use that if there is any hurry. They may not box it for you; you will have to buy the box but maybe also the packing materials and tape.
A service like UPS or FedEx will be far more expensive. But at a place like MailboxesEtc, they will box it all up for you and give you an all-the-way-through tracking number. I've used them, including from London, and it's expensive if you have books and/or a larger size. Expect at least $100 and more if it gets larger or heavier. $200-300 was my typical cost for heavier boxes 8 years ago.