We contracted with a shipping company to send mainly personal items (most of which were sealed in ordinary moving boxes) in shipping crates. We were given a range of quotes based on various combinations of 200 and 100 cubic foot crates. These are the standard sizes for smaller shipments. The next size up is a 20 foot long container which is suitable if you are moving a lot of furniture, but this size is expensive. Because we were paying for this move ourselves, we disposed of most of our large items, in a bid to save money by using only the smaller crates.
The key bit of advice I can give you is that the crates actually hold MUCH LESS than you might think. The numbers refer to the external dimensions of the crate. Internal size is in fact significantly smaller. In our case, I ruthlessly reduced our possessions to 165 cubic feet, reasoning that even allowing for some gaps between boxes, and considering that the internal dimensions would be less than 200 (the shipper was unable to provide me with the actual dimensions, unfortunately), our 165 cubic feet of stuff should fit into the 200 cubic foot crate with room to spare. I was wrong, and we were billed for 300 cubic feet. At this point there was nothing we could do, as the shipper already had our stuff, and there was no option to retrieve a few items and send them by mail instead.
In retrospect, I wish I hadn't disposed of so many of our things. I wish I had brought all of the stuff I was on the fence about. Our apartment here is large enough, and replacing the items that we sold or gave away is turning out to be extremely expensive. Plus, we gave away a lot of things that couldn't really be replaced (mementoes and such). And in the end, we had to pay to ship a nearly-empty crate anyway. Of course, each person's experience is different, and you will need to assess for yourself which of your possessions are trash and which are treasures.
If you have a lot of furniture, you may end up with the 20 foot container anyway. We did once use one of those for an international move, but the company paid for it so I don't know the cost. However, I can tell you that it was no problem fitting the contents of a one bedroom apartment (furniture as well as personal items) into the container.
I hope that your move goes well and that you find a solution that works well for you.