When we moved to Barcelona the first thing we took care of was finding a reliable and cheap removal company who also does international shipping.
The first option we tried was Anyvan since we used them before for our move from Germany to London. You basically put in where you want to move and how much stuff you have. Then you will get quite a few quotes and you can pick the best offer. The system also has ratings, so you kind of get a picture of who the best option is.
With this service you obviously need to know how much stuff you have. For us that was not really difficult, since we normally sell most furniture and only move with boxes. But starting to pack early really helps.
Some quotes will be really expensive and others quite cheap, but they all normally come with some sort of insurance. The removal companies also change their quotes based on other quotes you got and you can always ask them questions and try to get a different price.
In our case we also added our removal profile into Shiply to compare prices. That way we actually found our removal company.