Cleaning
Good old-fashioned soap and hot water is sufficient for most sex toys. Lather up the toy with a little squirt of unscented, anti-bacterial soap and let the foam sit there for a few moments to kill bacteria and other germs. Be sure to rinse well. If you're washing battery-operated toys that aren't waterproof, be sure to remove the batteries and avoid submerging them.
Toys made of plastic, jelly, or other porous materials, have to be cleaned more carefully, since germs love to sneak into tiny grooves that can form in these after a little wear and tear. Do the same as above, but with a little extra care to make sure you clean the surface as well as any grooves, which can be extremely small.
You can also invest in a sex toy cleaner. A specialized cleanser or spray not only kills off germs and prevents bacterial growth, but also uses conditioning properties which help to keep the integrity of softer materials like silicone and jelly.
Drying
To dry, you can either leave your toy to air dry on sheets of clean kitchen roll or speed up the process by drying with a clean towel.
Storage
Keep toys stored in a container or pouch (to keep them clean) and in a cool, dry place.
Note on lubes
Don't use silicone- or oil-based lubes with silicone toys, because it will stick to the material and break down the surface of your toy. Go with water-based or hybrid lube instead.