My just-turned-three-year-old is doing extremely well in a swimming pool. Here's what's we've been doing the past 2 month (we have a pool in our complex and have been using it since mid-April), maybe it will give you some ideas. She was comfortable in the pool with a swim ring since last summer.
First, I started with breathing. I showed her to take a breath and blow out her nose (exaggerated motions on my part). Not terribly exciting for her.
Mother's day weekend she went underwater for the first time. I held her facing me (close, her legs around my waist) and said take a breath. She said no a few times but I convinced her quickly before she could think about it much. She liked it and we did it a few more times. Keeping the momentum, I had her straighten her legs and touch the bottom, that was too fun for her.
Two weeks later we swam with friends, a 5 yr old girl (who swims well) had goggles on and we borrowed them, that seemed to seal it.. my dd's confidence/skills exploded. Add to that some sinking swim toys (in particular a treasure chest with jewels and coins that sunk..boy was that cool) and she was diving down, with help, go grab the jewels off the bottom (3-4').
The next time we swam with goggles and jewels, she was diving down with me in the deep end (5') to get the treasure. She holds her breath really well. I force rests so she doesn't gag/choke on water.
After the next visit with our friends, my dd would push off from me and swim in a circle and then back to me. She really wants to swim like the big girls.
She's learning to swim by me pushing her off toward the steps. It started with only a kick or two by her and now she does several kicks (more distance).
Of note-disclaimer-precaution:
-My dd is NOT strong enough to get her head above water, I LIFT HER HEAD UP OUT OF THE WATER (I believe age 5 is when they can). I am very comfortable reading her and know her limits and am always within arm's reach. It only takes 4-6 minutes of lack of air to cause severe brain damage or death.