Forget about using the built-in PWM in the microcontrollers (like the Atmega); it's too slow and produces flicker at low duty cycles. Plus you don't have enough channels anyway.
Look instead at TI's LED driver chips, like the TLC5940 I've been playing with. It gives you 16 channels of high-speed PWM per chip, and you can daisy-chain as many chips as you need. TI makes chips with even larger channel counts. These take serial input (and thus can be controlled from a cheap microcontroller with few pins) to which you basically write brightness values for each channel, and then say "go"! Super simple and effective.
You WILL need some kind of programmer hardware to program the microcontroller, no matter what it is. Bear mind mind that you can get those off eBay for like $3. An Arduino board is basically just an Atmega chip on a breakout board with programming interface (other than the "pro" versions which omit the programming interface to save space), preloaded with a handy bootloader. This combines with the invaluable Arduino libraries, which are just C libraries that handle a ton of everyday stuff. Don't dismiss Arduino; you can buy a Chinese clone Arduino Uno for little more than what you'd pay a Western supplier just for the microcontroller chip alone.
In a nutshell, I'd suggest you dip your toes into microcontroller programming and LED control the exact same way I have, since I'm working on precisely this kind of project:
- Cheap Arduino Uno boards (you can get 5 for under $15)
- TLC5940 chips
- IRLZ540 or similar logic level MOSFETs for high power switching (the non-logic-level IRFZ series is much cheaper, so keep a keen eye out on the part number)
- AVRISP II clone programmer (get one with switchable 3.3V/5V support)
- use the Arduino IDE to get the hang of the basics. Then move to a "real" IDE, but still using the Arduino libraries. (There are many options for every OS.)
- use pre-existing Arduino libraries for the TLC5940 chips
Now, I wouldn't be surprised if there are already "smart home" LED strip controllers that connect via USB, ethernet, wifi, or Bluetooth and can accomplish what you want to do. If this is a one-off project, they might not be a bad choice. Unless you want the challenge of DIY, of course!
Hope this helps!