A problem with getting moving in deep water is that there is no wall to push off from. And, if you don't get moving quickly, you are bound to get mowed over.
A simple trick to get from zero to race pace as quickly as possible is to go ahead and lay as prone as you can. Kick your legs gently until your whole body is parallel to the water surface and get ready to rock. Your competition will need to waste a couple seconds getting from vertical to prone, but you'll already be there and gone.
Next time you're in the pool, go to some deeper water and check out the difference. It's really cool!