Zerg Strategy: Timing the Opponent's First Push

I was recently playing a series of games against Terran and Protoss Very Hard computer opponents, and I noticed that as Zerg, you had to make Drones constantly until about 30 supply, and then spam either Zergling / Baneling (vs Terran) or Zergling / Roach (vs Protoss) or I'd lose horribly to the computer's massive push.  Occasionally, the Protoss opponent would push slightly later, adding Immortals or perhaps a Colossus.  Making an excess of Roaches to defend this second push would render me very, very dead, even with 2 Spine Crawlers, unless I built a ton of zerglings to rip apart the Immortals and Stalkers.

So... figuring out when your opponent's first push will likely come, and with what units, will allow you to defend it without drones, and if you do this, you will be ahead in economy, and even a simple double-larvae spawn + all military units might let you counter-push and win the game, even on low-tech.