Is there any way to ship my belongings to the UK if I decide to move there?

Unless you find employment with a company that will pay to ship your things, it really does not make a lot of sense to ship a lot of things to the UK from the US. It might make more sense to store the stuff you really want to keep. I don't know how old you are, but as you get older you may find that stuff is not that important. The cheapest way to ship your things would be surface shipping - by sea. Check with any shipping/moving company and they will be able to point you in the right direction., even if they don't actually do that type of shipping themselves.