It really depends on where you want to go with it. I found early on that I learned most by figuring out how to achieve self contained blocks of functionality to accomplish a more elaborate task. For example, to build a tape following robot you need to run the motors, read some IR sensors, implement some kind of PID control, build some kind of regulated power supply, etc. (I learned least from solving circuit problems out of a textbook).

Each of those tasks can pretty well be accomplished by building standalone circuits, none of which are particularly complicated. In the process of building and debugging each one, you'll find you need to figure out how some of the more basic building blocks of electronics work, for example, capacitors, LEDs, opamps, transistors. There are a huge amount of resources out there to learn from. For anything related to arduinos or microcontrollers, just google it. I don't mean to sound condescending, it really is something that can be googled very easily with relatively high quality results (unlike some other areas of electronics, such as analog amplifier design). For some more theoretical stuff, I suggest you check out MIT OpenCourseware as well as video lectures from Standford on youtube. I blundered my way through a physics degree and an electrical engineering degree using those sites. For a more gentle introduction to some electronics concepts and applications, maybe check out this: http://electronics.wisc-online.com/Category.aspx?ID=3

Now the bad news. You can do a lot of futzing around with arduinos and leds, and that's great, but you'll soon find out that it's interaction with sensors and actuators that makes electronics interesting. To do that you will find that you need to build/use external 'hard' circuits, and to do that you will eventually need an oscilloscope. You really can't get too far in the field without one, even if it's just a hacked sound card and audio cable. I say 'bad news' because a new benchtop scope is pretty expensive. A breadboard, some wires and a soldering iron don't hurt either ;-) For components and data sheets, digikey.com and octopart.com are your friends. For breakout boards and other hobbyist accessories, sparkfun.org and adafruit.com are your friends. There's also a arduino plugin for sublimetext, in case you're like me and can't stand using that atrocious java UI.