I want to start a rock band, and if possible play in my county fair's battle of the bands, but I really don't know where to begin.

I play guitar, my brother plays bass, and I have dozens of friends who play guitar and piano.

The problem is, I don't know any drummers, and I don't know anyone who can sing decent (I would like to learn, but I don't have the money for lessons).

I also have no clue what kind of PA system, microphones, and other stuff I would need.

I also have experience playing with other people, as I am currently in a church band.

You have two options:

1) Talk around to see if any of your friends know drummers.
2) Place an ad or look for ads online or in a local guitar shop.
Only play what you hear. If you don’t hear anything, don’t play anything.
-Chick Corea
If your lucky you might be able to find a singer with a PA and microphones. After all, guitarists, bassists and drummers are usually expected to own their own equipment, so why shouldn't singers? With regards to finding a singer, if your still in high school, maybe try asking people that sing in the school choir. And a school concert band or marching band might have some drummers? Asking friends can work well, but you already said none of your friends sing or play drums. But have you asked your friends about whether their friends, siblings, cousins etc. can sing or play drums? If none of that works, you'll just have to look at ads/place an ad online, in a guitar-shop, in a record store or around your school or university.
Start with what you have. Most bands start with a core of a couple of mates playing together with other people joining in and dropping in and out of the band only later does this coalesce into a permanent line up. Find somewhere you and your mates/brother can meet and start making music together. Preferably meet regularly (once a week?) and start learning some songs. You can all try singing, lessons are great but so is practice.

Let anyone who plays know what you are up to and invite them along to join in. Most of all it should be fun, the joy of making music together but at some stage you'll find you are close to having a band forming and can start to look at open mic's and band contests.

If you stay welcoming and stick at it then other musicians will want to join you as you'll be the ones actually doing something.

Good Luck
My band took 6 months to form, I've gone through 10+ members trying to find the perfect line-up. Finally managed to get a really good group of lads and a permanent line-up.

Make sure your acquaintances know you're looking for a band. One of me mates rang me up at college telling me he'd found a bassist who might work. That same bassist is in the band and doing more than you'd expect.