You trust the moving company to manage your belongings properly, and with care, to prevent damage. However, in some situations, damage happens. If you can prove the company was reckless and negligent, you may be able to sue the company to get back some of the loss you are dealing with. How likely you are to win such a case depends on state laws and the circumstances of the situation though.
The first thing to do is to document the damage. Take lots of photos and describe the outcome of the move. You also need to have the full names of the movers to ensure you can prove they were there. Know the worth of the damage, either based on sales receipts or estimates on replacement value.
Read through the contract you signed with the company. If there are any types of clauses that limit responsibility for damages, note them. Most moving companies have wording within the contract that releases responsibility for damage that is unavoidable or that occurs when the company is taking great care to protect your items. You will need to show that the employees of the moving company did not do this to win your suit.
If you believe the movers were negligent, contact your attorney to file a lawsuit. You can sue in small claims court without an attorney, but having an attorney by your side for larger lawsuits can help you. Otherwise, visit the local small claims court and file a complaint. You can then request a hearing to determine responsibility. The process could lead to the company having to pay you for the damage.