As far as fabric choice goes, putting satin on shoes was a terribly idea. The lovely shiny finish is gone as soon the shoe touches dirt, and satin spots when it gets wet. The best option to keep your shoes pretty is to wear them as little as possible, and that can get in the way of practicing in them and improving your dancing.
While I know of no way to restore the shine of new satin, you can clean your shoes to improve their appearance. Perhaps you want to get a big scuff mark off before a special lesson or workshop. To do so, mix up some room-temperature water and your fabric cleaning agent of choice. A clothing detergent that can be used for hand-washing would work great. Using a soft cloth (something that doesn’t have much of a ply, so not a dense terrycloth) wet in your soapy water, wipe off the dirt using circular motions. I would normally wash the entire front half of my shoes this way, to avoid any extremely obvious line as to wear my washing ended.
Immediately afterwards, grab your hair blow-dryer and dry your shoes. You don’t have to get them bone-dry, but getting most of the water out will prevent the satin from spotting. The shine will be gone, but your shoes will be clean.