Building an RGB LED module
Look at WS2812 or WS2812B. 

They are an RGB LED in a 5x5mm package.

As well as a red, green and blue LED, each one contains the PWM circuit to control brightness of each colour. Give it power (straight from a power supply), and it'll do the rest.

Further it handles one-wire serial data input and output. They handle data at a rate of 800kbits/second.

So it is possible to wire hundreds in series into one 'string', talk to the string with a single wire from e.g. an MCU, and change all of them in less than the refresh time of a TV picture (i.e. under 1/100th of a second).

It handles making the correct colour, using its built-in PWM, without any further processing. It will carry on showing the same colour as long as it is powered. 

So you only need to provide power to it, and send data to it when you want it to change colour.

You can buy small numbers of them ready assembled so that you can practice, and become confident that you understand how to use them for a modest amounts of money. 

For example 8 RGB pixels, though there are lots of products and vendors. You can buy individual components for between 10¢ and 20¢ from China, as well as flexible strips, panels etc.