Thank you all so much for your helpful advice! I really appreciate it. 
As it turns out, my father had this wonderful textbook in his library about electronics, and it goes into great detail about wiring diagrams, schematics, the relationships between voltage, current, and resistance (i.e. Ohm's Law), and a bunch of other stuff I haven't read yet. I've been reading it and doing the problems to help familiarize myself with the kinds of things I can expect, and it's turning out to be a lot of fun. I'm hoping to try applying this new knowledge soon on some simple combinations of my leds, resistors, and my Lilypad. I'll move onto the button and multiple LEDs later.
Referring to the LEDs in the Arduino IDE isn't at all what concerns me. What concerns me is making sure I connect the proper power supply to all of this (in the right way) and make the connections to all the components without damaging anything. I've never soldered anything before in my life, and I'm scared to death I'll fry the circuit with a cheap soldering iron or something. I don't want to have to buy a new Lily pad. :x
And yes, I realize that a LilyPad is WAY overkill for a simple project like this, but try to see it from my own perspective. 
coding and software interfaces are in my comfort zone. Hardware components are not. When I bought my components, I bought the ones I thought would be the bare minimum I would need to deal with in order to achieve a workable solution. It's like liudr said; program logic is easier for me to deal with than hardware logic, so unfortunately, that's what I bought into. At this point it's already a done deal, anyway. I've already bought everything, and I want to give it a shot.
Thanks a bunch to all of you! I'll give this a good shot, and hopefully I'll have some good news to talk about later. :3