I am currently developing a lighting system using Arduino that uses 12V LED strips. To control each colour, I have seen that you have to use TIP120 transistors. Can I substitute these with 2N2222A transistors, or can these not handle the power?
Probably not. What's the current rating on the LED strip?
The 2N2222 is only rated to 600mA (and you might need a heatsink and you should leave some safety margin). The TIP120 may also need a heatsink.
...Worst case, you'll burn out the transistor so if you have extra 2N2222s you can try it. Generally, transistors short-out when burned-up so the LEDs would be stuck-on.