I want to learn basic electronics/circuitry

Perpetual

What are some ways I can learn basics of circuit-building and stuff? I heard there are "kits" you can buy at RadioShack and stuff that come with manuals. Are they legit?

Basically I want to play with soldering, LEDs, and eventually learn enough so that I can build an amplifier for my stereo system.

(I have no interest in taking formal classes or pursuing a degree.)

Fantasma

You may find many used books at amazon.com, electronics is an expensive hobby, and to a certain way challenging. You must also consider your own safety, you will be handling electricity and chemicals that with time could potentially cause cancer.

RadioShack sells good stuff, but quite expensive from my viewpoint. You said you don't want to take classes, however, there used to be good correspondence courses.

Coinage

For learning principles, these very long PDFs probably contain everything you would want to know, and you already paid for them with your tax dollars.

For parts and stuff you're better off going online than Radioshack. Sparkfun and Ada Fruit might be able to help you there.

mightyjongyo

Sparkfun sells some kits with instructions. They have one that basically just blinks LEDs at different rates, but it teaches you through-hole soldering, so that would be a good start as far as that goes.

Build an amplifier or design an amplifier? The latter will be hard without formal coursework, I'd imagine.