There are actually an amazing number of different styles correct ways to use the chopsticks. Here are the 2 basic ideas, which lead to other variations:
The immobile and mobile pair chopstick method:
Holding a chopstick while eating.
Hold the first stick stationary between the base of the thumb and the middle finger. (The thick end is held, not the thin end).
Hold the other stick in with the top of your thumb and the tip of the index finger. Even up the pairs.
Move the 2nd stick while holding the the 1st chopstick stationary.
Using the other stick as a pincher, and pick up your food in this way.
Variation in the immobile pair chopstick method:
Hold the first stick stationary between the base of the thumb and the middle finger, but with the tip of the index finger also touching the chopstick. So the 1st stick is sandwiched in the middle of the index and middle finger.
The other stick is held between the thumb and index finger.
The two are moved somewhat together, in more of a pincher movement.
Eating rice this way is actually not too difficult. For starters, most Asian rice is "sticky" so it is easier to grab clumps of it. Also, use the pair as a sort of scoop to gather up larger clumps of rice from your bowl.
Chopstick Usage & Etiquette
When not in use, lay them down in front of you with the tip to left.
In Japan, before you start eating Say "ita-daki-masu." It means basically "thank you, I am going to eat."
When you finish eating say "go-chi-so sama-deshita." This means "thanks for the food, it was delicious." Japanese say both these things so often that they almost don't think of the actual meaning.
Normal chopstick technique
Alternate chopstick technique
Slurping soup is not considered rude in Japan.
If there are bowls of common food, you may use the thick end of the set to get food from the main food container to your own bowl.
But not this…
Placing your chopsticks in the rice bowl standing up is taboo - reserved for the departed after a death, in memory of the departed loved one.
Avoid transferring food from one chopstick set to another, which is another custom involved with a recent death.
Don't spear or stick food. Not good manners.
Don't point with these utensils to something or somebody. Same as pointing a knife or fork at a guest.
Don't move or wave your chopsticks around in the air or above the food, same as a fork and knife.
Try to avoid clanking or scraping on your mouth and teeth. Likewise it's considered rude to suck on the ends of the sticks, even if you love the food.
When serving yourself food, always use shared serving chopsticks or spoons if available as opposed to digging in with your own chopsticks. This helps avoid the spread of disease as well. This rule is sometimes not observed in China however, and so those unacquainted with Chinese customs may be surprised.
For hygiene's sake, when obtaining food from the serving dish, the chopsticks are inverted the other ends to pick up the food.
After you have picked up an item, it is yours. You should not put it back in the dish. Therefore try to decide what piece you really want to eat before grabbing it with your chopsticks.
Sometimes you may want to just pick a choice piece of food for your guest and serve it to them. Be careful though that they are not on a special diet, and be sure they will actually like the food before serving it with a shared chopstick or utensil. As mentioned before, never stick the set into a bowl of rice pointing up, as this is done to remember the recently departed dead in families morning the loss of a member.
As expected, chopsticks are not immune from the realm of superstition. If you are given a pair of chopsticks with uneven lengths, it may indicate you are about to miss your boat or airplane. Dropping a pair of chopsticks is like stepping on a crack in the sidewalk. Not a good idea. Amazingly, with all the of the do's and don'ts about chopsticks, you ARE allowed to place them on the table crossed!