Brewing Kombucha is insanely easy. Way, way easier than beer or anything else.
1. Buy a large glass dispenser with a stainless steel spigot at the bottom (this is important and will make life WAY easier).
2. Buy a Kombucha starter kit online or locally. I usually use green tea as my base. I buy my kits from Amazon and they come with the scoby (mother) and tea bases. Some come with glass jars, etc. I'm looking at them now and there are a LOT more options than when I got into it a few years ago.
3. Cover the top with cheesecloth so it can breathe. Make sure mold stays out. If you get mold toss the batch and start over. As the scobys grow and break apart and you drink (using the spigot at the bottom) they float to the top, with the bottom spigot you can easily pour without having to move the scoby out of the way.
I haven't brewed it in a few years, but my first batch ever wound up getting mold in it. I'm not really sure why, or how, the rest of my batches were fine.
It was very obvious, it looks like just the white/green mold that will grow on top of a drink if you leave it out.
edit: Oh, the scoby is always obvious, it's basically a gelationous alien looking thing that should star in Xfiles. Every few weeks the scoby will split off and you'll have multiples, you take out the second one and can give it to someone else or make a new batch.