Here are some games to motivate your child to put their face in the water. You can use these same games when practicing other skills too — just substitute "face in the water" for "front float" or "back float," and so on.
For these activities, you'll need torpedoes, rings, and a treasure chest. You can also create your own games — keep it simple.
Don't overdo it. Somewhere between three and ten attempts at most, depending on the child. Remember to follow their lead and stop when they've had enough.
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November 2018 update: I originally wrote this post in 2016. In my continued research to find new ways to fix water anxiety, I discovered these activities — and they happen to be extremely effective for the 2-year-old+ child too. You'll see how easy it is to get your kiddo to put his face in the water, even without goggles and a nose clip.
Torpedoes

I like to show my students how the torpedoes move swiftly through the water. You're going to use them to motivate your child to put his mouth in the water.
Tell him that when he puts his mouth in the water, you'll give him a torpedo to throw himself — give it to him even if he just puts his mouth in for a split second.
Do it again. This time, challenge him to keep his mouth in until you count to one, then he gets another torpedo. Continue until he can keep his mouth in for 2 or 3 seconds. Follow the same approach for the nose, and finally the whole face.
Rings

Hold the rings behind your back. When he follows your instructions, have him guess which color ring you're going to pull out — then give it to him regardless of the color.
Treasure chest

There are many options for kids' treasure chests. I use the Melissa & Doug Sunny Patch Undersea Treasure Hunt — I love it because it comes with six items (four coins and two gems) and has an opening at the top for the coins, like a piggy bank.
Have your child follow the instructions from the torpedo game. When he does, hand him a coin or gem to put back in the chest. The coins are the best, because they get to drop them through the slot. Turn it into a guessing game.
These are simple games and great motivators — my students love them.
Now that he can put his face in the water, the real fun begins
When they get comfortable putting their face in the water, they'll be able to reach down to the bottom in the shallow end to pick up objects — and finally dive down for them. When they're ready for this, start with the rings; they're easier to grasp.
Want a head start? Get my free course — How to Prepare Your Baby for Swim Lessons (0–8 months).