Hey there. I'm just getting started as well and, like you, I started with a solar powered LED device. In my case I was attempting to build one of these Sun Jars without just cramming a solar garden light into a different container. I studied a few circuits and settled on something like the one found here but I also did some research into the different components and learned how to read circuit diagrams and figure out how everything works together.
One site that helps a lot was the electronics section of Technology Student, there you can learn what each symbol in a circuit diagram is and what the components do. Also, a good and inexpensive book is Getting Started in Electronics by Forest Mims.
If you want to learn more about solar cells while making cool stuff check out BEAM robotics, **broken link removed** is a nice start.