Take a look at [u]this thread[/u].
Can you help me on what hardware/software that is best to use...
...and also would like to try at first using an 8 channel relay module and then upgrade it to a larger setup.
Mechanical relays are NOT the best for rapidly blinking/flashing lights. For LED strips a MOSFET (one per channel) is probably best.
Also, with a MOSFET you can dim the LEDs. If you want to dim, you pretty-much need one Arduino output for each channel. If you don't need dimming you can have almost an unlimited number of channels by using shift registers. (I've built 3 sound-controlled lighting effects and none of them use dimming.)
With multiple separately-controllable RGB strips the number of channels can get BIG quickly, and that means lots of MOSFETs and lots of wiring. i.e. 4 separately controlled RGB strips requires 12 channels. An 8 x 8 RGB matrix is 192 channels! (An 8 x 8 single-color matrix can be wired as an actual matrix requiring only 16 channels, but the common cathode or common anode on RGB LEDs means they can't be wired as a matrix.)
It's up to you, but you may just try 3 channels for the 3 colors... One of my effects is a 4-channel AC controller and depending on how I'm using it, I control 4 different colors of flood lights and I plug-in 2 red, 2 blue, 3 green, and 2 amber floods. And of course the color-pairs are plugged in together so both reds come on at the same time, etc. In another configuration I use 16 smaller white floods with the 4-channels arranged in a line, square, or circle for chasing/sequencing effects.
I haven't done anything with RGB lights... yet. But, I'm thinking about an RGB project and it would have 2 RGB channels (6 channels). My idea is the two RGB "stage lights" could operate together, or they could be "complimentary"... so for example, they would be flashing between red and green and they could be "opposite colors" (there would also be a mode where they are both doing the same thing, so effectively only 3-channels.) And, there would probably be a mode where the left light responds to the left channel music and the right light responds to the right channel music.
If I do this stage light project, I'll be using [u]DMX control[/u] which uses addressable serial control and the wiring will be easy. Since DMX is addressable, you can have many-many channels with one physical output connection. (But, I don't have the budget for lots of DMX lights.)
On the software side, you should be able to find some examples, but I'd just recommend you start with the blink LED example and blink without LED example, and then add more LEDs/channels and be creative to do whatever you want. (You'll generally need to follow the blink without delay example because if you use delay, your program/sketch can't read the audio or do anything else during the delay time.)