I am meeting with customers all the time, and they always ask, " What can I do to keep the cost down?"
The best piece of information I can give to them is to go through their house, and if there are items that they haven't used in the past year, then it is probalby something that you can part with. Local moves are billed on an hourly rate, and long distance move charges are based on weight. By limiting what they move they will save on time needed to perform their move, or weight of the shipment.
The process of going through your posessions should take place before you have an estimator come to your home, that way you have an accurate description of what is moving and you can get a more accurate estimate.
Also, by doing this early in the move planning stage you give your self time to find charities to donate the items to, or sell the items at a tag sale, craigslist, or auctioneer. It does not make sense to pay to move items that you no longer have any use for.