Is getting a kit like this the recommended way to learn electronics? I don't know anything about it! I would like to get to the point where I can light up a LED bulb programmatically and understand how it's happening.
There is no one way to learn electronics. The Arduino will hold your hand through lighting up an LED, but depending on how much depth you want, may not teach you how it's happening. Working with an Arduino is like bowling with bumpers, which is a good place to start.
You could probably spend less and learn more about electronics by buying a few key components like a transistor, a LED, a 9v battery, a switch, a few resistors, a few capacitors and some insulated wire. You don't even need a breadboard--you can tape or clip things together just to see what's happening. There are tons of simple circuits and tutorials for just these components.
