Rubber is degraded by excessive heat and UV light, so it should be kept stored in a cool, dry, dark place. Rubber clothing is best stored hung up on plastic hangers (Not metal, some metals also degrade rubber) or failing that dusted with talc and rolled up.
If you need to clean your rubber gear it should be cleaned by washing it in tepid (not hot) water and a mild non-greasy soap, don't use strong strong detergents or ones with added moisturisers, as the grease in them can harm the rubber. After washing the rubber it should be thoroughly rinsed to remove all traces of soap and then hung up to dry completely before storage. Adding a small quantity of talc to the rinse water may help in preventing the rubber sticking to it's self as it dries.
How you put on your rubber clothing will of course depend on what it is, thick industrial rubber like dry-suits will take a lot of abuse and only the most basic common sense precautions are needed. Avoid sharp jewellery, and be careful not to allow your fingernails to dig into it when putting it on. Thinner rubber will need more care to get on, you may need lubrication to avoid tearing it when pulling it on. Water based lubricants or talc are suitable, oil based lubricants (including most hand creams or oils) are NOT and will rapidly damage rubber. You may find that it's easiest to roll up the clothing before you put it on and then unroll it onto your body.
To get a good shine on your rubber clothing, wipe away any excess talc or lubricant with a slightly damp cloth, and then silicone spray can be used with a fresh dry cloth to put a high shine on the rubber.