What is the best way to clean my Apple Watch?

According to the Apple Watch User Guide, the Watch should be dried and exposed to fresh water after "workouts or heavy sweating". The Watch should not be exposed to soap, detergent, acids or acidic foods, salt or pool water, insect repellent and other cleaning products or abrasive materials. If your Watch is accidentally exposed to any of these materials turn it off and wipe it down with a slightly damp cloth. A "nonabrasive, lint-free" cloth is also recommended for keeping your Watch clean.

Remember, the Apple Watch is water resistant, but not waterproof. This means it should not be submerged in water but it can withstand things like rain and sweat. The Apple Watch should never be in water for long periods of time, and it should not be cleaned while powered on or charging, or dried using an external heat source.

Update: The official Apple Watch manual notes that the oil repellent coating of the Apple Watch will wear over time. One way to avoid this is to buy a screen-protector, like the Zagg InvisibleShield, to protect your display from wearing down.