Locate equipment or moving rental companies near you that rent enclosed utility trailers. You might find it useful to call several to compare rates. Try to call all your local companies that might not have an internet presence. Compare these to the national moving truck and trailer rental companies' web-generated rates.
Determine the size of trailer you need and the towing capacity of the vehicle you'll be using to pull it. Some truck and trailer rental companies' websites have packing-space calculators that determine trailer size according to how much furniture you have. Contact your vehicle manufacturer if you can't locate your vehicle's towing capacity, or ask personnel at the rental company whether your vehicle will be able to pull a trailer as heavy as the one you're looking to rent.
Go to the rental company you've decide on and rent the trailer. Most companies will require you to provide a major credit card at the time of rental. If you don't have a credit card some companies may accept a refundable cash deposit.