As far as the elemental points go, they do a couple of things.

First, they apparently effect the rates at which you get certain types of crystals. If you've got a ton of points in fire, for example, you'll supposedly get more fire shards as drops/rewards. (I'm not exactly sure how this works, though.)

Second, they effect your ability to do magic of that type. So if you put points in fire, you can do more fire damage on fire spells. (It gets a little more in depth when it comes to astral/umbral magic. Since you're just starting out, I'll leave that for another time.)

Third, they effect your magic defense. But this one's a bit different. FFXIV uses an elemental wheel. (link to a picture of the elemental wheel) If you put points in fire, it increases your defense against ice attacks. If you put points in ice, it increases your defense against wind attacks. If you put points in wind, it increases your defense against fire attacks.

Oh, and a little piece of side advice. Always listen to Pang Tong. That dude is awesome.