Understand transistors
Since you have a good grasp on the other components, transistors should be no problem at all to understand. Doing a quick search around here, I found a post that I think sums it up very well.

Basics of Transistors


  Think of a NPN transistor this way: You put a little current thru B-E, and that allows a lot of current thru C-E. The ratio of a lot to a little is the transistor gain, sometimes known as beta and sometimes hFE.


To sum it up, a common use of transistors is as an amplifier, or even simpler, a switch. 
A good example would be powering a motor by using a microcontroller for a robot. You want to be able to turn the motors on/off, which is what the microcontroller will do. If you were to hook the motor straight up to a MCU digital pin, you would destroy the MCU because it cannot handle the currents typically needed. Instead, you use a transistor that will use a small amount of current through the B-E but will allow larger currents to flow through C-E.