I recently moved and did what the previous poster recommended - taking extra bags on the plane - and I also used the Post Office to ship items, such as books, clothing and shoes. They usually offer the best rates for international shipping. I sent about six boxes (approx 30x30x60cm) and they went by the size, not weight, so I used compression sacks (found at outdoor/camping stores) to put clothing in and compressed them down to get as much clothing as possible in the boxes. I used the Royal Mail, but I'm sure USPS offers similar services. My stuff also only took 3 or 4 days to arrive, so I wasn't waiting weeks for everything to arrive, which was very nice.
Check your airline's excess baggage policy to figure out how much it will cost you to carry excess and/or overweight bags, versus how much it would be to ship. You can usually get an accurate quote for shipping online by entering dimensions and weight of the boxes. We flew Air Canada and took as many bags as we could (one free bag plus two extra bags per person) and had two that were also overweight. The woman at Heathrow check-in only charged us for two excess bags and two overweight bags, instead of four excess bags plus two overweight fees. I think it was somewhere around £200, plus £300 for the boxes to be shipped. We also did the maximum carry-on bags, which was one large carry-on, a backpack and purse or messenger bag. It depends on the person checking you in, because I had four bags on United when I moved to the UK, and wasn't charged anything. They're exceptionally nice at Denver, though.
The airlines also offer shipping. You can take your boxes and/or bags to the airport and pick them up from the airport when you arrive. For me, it was around the same price, so I opted for the Royal Mail option, so everything would be delivered to me instead of having to pick everything up.
Best of luck to you with your move!