The key is finding a container line that uses a port that works on both ends. The last time we went from the UK to the USA, we used Atlantic Container Line (ACL), which went from Liverpool to Norfolk, VA. We have also used Sea-Land (now defunct, I think) and Maersk.
Find a shipping agent - on this side, think Liverpool or Southampton. In the USA, if it's the midwest think St. Louis, New Orleans, Mobile - because some of the haulage might well be river. The road transport can add a lot, but what adds the most is going through a 'moving company'. A good shipping agent will arrange everything and guide you through the entire process. Some container lines will deal with you directly, but some won't as they know that many folks don't understand what's involved. We've shipped other stuff so DH knew the procedure.
BTW, we have never bothered with the insurance. Not costly, but if you read the fine print, you won't have much of a payout if the 'worst' happens.