Okay, I'm a textile designer in the fashion industry and here's what we learned waaaaayyyy back in my college textiles classes:
You can totally do this yourself. Crazy as it sounds, a lot of drycleaners will simply throw a wedding dress into a washing machine to "professionally clean" it! Ha! They will undoubtedly follow up with steaming the wrinkles out of it, though. And then they fold it nicely with acid-free paper for you to store away in an acid-free box for whatever future use you may anticipate.
Because you probably don't want to have to do the steaming part, I'd skip the washing machines, though. They will wrinkle the heck out of the fabric and then you will have to try to iron or steam it yourself and that's a pain. Hand wash it and let it drip dry (no wringing!) You may have to touch it up with an iron but, it will be much easier than trying to get all of those spin-cycle wrinkles out.
Your gown is satin but, the important information to know is wether it made of silk or polyester or what? That is the real way to find out how to care for the fabric. No matter what the fiber content, you can't really go wrong with hand washing it and letting it drip-dry on a line or on a hanger in your shower. There are a couple of variations though, depending on the material.
If it's silk, wash it in a sink or wash tub using cold water with Woolite or Ivory dish liquid (the Ivory's actually really gentile and that's how I wash my fine hand-washables in lieu of drycleaning them. i.e. cashmere, silk, etc.). If your dress is polyester, just hand wash it with a conventional machine detergent like Tide and spot-treat the really dirty parts.
For the dirty spots: wet the stains with warm water (cool water, of course, if it's silk) and pour a little of the soap or detergent on the spot -- enough to cover the whole stain but, you don't have to go over-board. Then rub the fabric together with the soap to work the soap in to the fabric and the dirt out. Rinse, repeat if necessary.
Best wishes with the project! If you have any other questions, there's always Google!