I would maybe ask the bridal store that you bought your dress from if they have any recommendations. Also post in your local wedding sell/swap groups on Facebook. You might be able to find a bride that has had great experience with a certain company.
Call up a few dry cleaners in the area and ask about their processes and prices. That will give you a good idea on expectations. You can also ask the store you bought the dress from for any recommendations as they likely would have a few local contacts.