Take it slow, and make sure you do a good job.

• With a cotton ball, run oil around your hairline and onto the tips of your ears. This will keep any color that bleeds or splatters from dying your skin.
• Wear a shirt that you hate. You're going to splatter, might as well be prepared.
• After you are done applying the color, pile your hair on top of your head, and leave it alone for the allotted time. If you have a lot of hair, pop a shower cap on it to keep it under control. If you don't cover it, be careful not to lean your head against anything, like a pillow or chair.
• After you get your hair under control, hunt around the area where you did the application with a wet rag. Wipe up any splatters. They won't be permanent at this point, and will come right up. I always apply in front of the bathroom sink because I can see what I'm doing and all the surfaces are easy to clean.
• Rinsing. Put a handful or two of water into your hair, and scrub it in. This will loosen up your hair and let it fall down if you've piled it up. It will also cut down on the splattering. Rinse your hair until the water runs clear. Don't forget the sides, plug your ear and let the water hit the side of your head.
• After the rinse is done, your hair will feel dry and rough. Dump a lot of the conditioner that came with the dye on it, and work it in really well. There is no such thing as too much, your hair will soak up a surprising amount. I have short hair and use about 1/3 of the bottle. Leave it in as long as the directions say. Do this every week or two as your hair gradually dries out.
• Dyed hair is thicker and drier than undyed hair. Condition whenever it feels dry. Your blowdryer can make the dryness worse. Always use conditioner or some sort of product to coat your hair with protection before you blowdry it.