I'm familiar with the general mechanics of the game, but what's the best build order for the beginning of the game?
An opening dictates how you are going to start the match.
The basic Zerg openings:
How to read these: The number indicates the number of drones you should have before your first structure. Since you can only build 10 before an overlord any number over 10 indicates building an overlord on 9 or 10 (9 or Extractor trick is the superior choice). Pool indicates building a Spawning Pool. Hatch indicates building a Hatchery. Gas, indicates building a Vespene Extractor on your nearby Vespene Geyser.
Which opening you choose usually relies on two things:
The Rush: A Rush opening is necessarily an all in, so that is a decision you are making prior to starting the match.
Standard: 13–14 pool is considered standard play, it can easily lead to fast expo or 1 base or speedlings or any of another things. It's considered standard play because of the options it offers you. I believe Slush of Root Gaming still runs this build.
Speedling: This is a strat that cropped up near the end of Beta mostly on Korean servers. If you build your gas before your spawning pool and immediately throw 3 drones in it on completion, then you will have exactly 100 gas when your spawning pool finishes. This leads to early speedlings for heavy pressure. Artosis is a large proponent of this build.
Macro and Macro+: Both of these are fast expansion builds designed to get up a very good early economy. They are very vulnerable to rushes, and require good scouting to support. The advantage of these is that if you can scout when your opponent leaves his base you can quickly put together a large force that will be ready just as he arrives allowing you to hold off the assault and be in better economic position. Idra and Machine tend to run this build.
Sen runs a 12 pool to give him a larger advantage against more aggressive zerg opponents, which are common on the asian server. Outside of him, it is not a very popular build.
9 Overlord vs 10 Overlord
There has been a lot of debate as to getting your overlord after your 9th drone or after your 10th drones. To add confusion it is possible to get your overlord after your 11th drones by building a vespene extractor, building the 11th drone and then canceling the extractor to get your drone back. It is usually agreed that getting your Overlord on your 9th drone, or building it after your 10th and then using the extractor trick to get an 11th while your overlord is building are the better choices.
There is some evidence to indicate 9 overlord is slightly better.
I first wrote this post shortly after beta and while most of what I wrote here still rings true there has been a shift in the Meta game in that time. While 13–14 Pool is still an effective strategy, many more Zerg have shifted towards more Macro oriented builds. 14–15 Hatch in particular has really caught on in ZvT. Everything in the post remains valid but I think it's still worthwhile to update on some of the newer developments.
Lately I've been using the Overpool build not described here:
Using the drone from the extractor trick to scout and harass, I can respond accordingly quickly when the pool finish. Make spine crawler to defend if there's early pressure. build zerglings at 11 to harrass Fast expansion, or drone up as if it were a 13 pool build.
The Overpool is also the route to doing the quickest 7 Roach Rush.
i don't know where 13 pool came from but 14 pool is standard, not 13.
i find the best opening is 14 pool 15 gas 15 overlord 16 roach warren/queen. you can build up to 6 zerglings if your scouting finds something dangerous or spinecrawlers if you find some sort of allin attack coming, and a quick 5-7 roaches will let you get early map control and choose whether to expand or tech. usually expanding then teching to hydras is the right answer, depending on what your opponent does.
Just to add, you can build the pool after 6 drones if you want to do a very fast rush. I dislike this approach, but have seen players use it.