Simple circuit to fade in / fade out LEDs randomly
For fading you'll probably want varying PWM. Generally you'd want to do this with microcontrollers. The Arduino is a very simple platform that is cheap and easy to use. I don't recommend it if you really want to learn how microcontrollers work, but if you just need something to get work done, it's definitely your best option. It has built-in PWM functionality, as well as MATH functions (like a random-number generator, which can be used to determine which LED to control), and the Uno (probably the most common model) has 6 PWM pins. You could also go for the Mega if you want more PWM outputs. 

You could also do this with multiple 555 timers (one per LED), but this seems like a lot more work than simply using an Arduino. However, if this interests you, here's the schematic:



This site explains it a bit more: http://www.555-timer-circuits.com/up-down-fading-led.html

You can change the capacitor value to adjust the fading speed. By offsetting this speed slightly, it will appear that the LEDs are fading in and out randomly.