Here's the thing about advanced stats, you can't just learn a few and use them the way you used to judge guys based on batting average or ERA. It's a rabbit hole, where different stats relate to each other and tell you different things and you have to understand what they are evaluating to be able to use them in relation with each other.
The primary advanced stats that try to simplify or rate a player in a single number, like OPS or WAR, or FIP are not really valuable for anything other than as a very loose comparison metric. They're ways to try and say which kind of power hitter is equal to this kind of defensive middle infielder or how a high velocity flamethrower contributes about as much as a ground ball control specialist.
It's fun to be able to say these guys are roughly equal or show why you want one over another, but where advanced stats are actually advanced, from a fan's perspective, are when you find something and learn something just off numbers. You can look at variance in a player's performance, look at his BABIP, HF:FB%, LD%, his OSwing and ZSwing rates and you'll start to see a picture of what the player more likely will accomplish in the future. The same kind of things for pitchers.
My main point here is that OPS or WAR or FIP are "advanced" stats, but they're advanced stats trying to accomplish what we mistakenly used BA, RBI, R, and HR for in the past, to calculate a player's totality.
Here's where I'd start:
http://www.tangotiger.net/runscreated.html Things have moved on from here quite a ways, but I think it'll give you a clear sense of what "advanced" stats are trying to do. Once you understand the motivation it becomes easier to appreciate why it's worth the effort.
Or, for other reading or just to catch you up on what a particular abbreviation means, the Fangraphs glossary is solid:
http://www.fangraphs.com/library/The