Here's a possible solution as an attachment. I had to mask some of your schematic and was not able to get rid of the black outlines.
Q1, 3 and 5 should be PNP not NPN Transistors, especially if V+ is greater than the V+ of your Arduino.
You will now have to pull the Base on each of those LOW to turn on those Transistors.
Q2, 4 and 6 where wired completely wrong because the emitters should be going to ground.
Here's a modified schematic and an alternative, better way to do this if these are individual and not RGB Led's. R10 - 18 are 10 - 22 Ohm Resisters that help compensate for the slight forward voltage difference that Led's have. These will help to make sure each LED is much closer to the same brightness when in parallel.
The section starting with Q10 is a modified version that places the resisters on the anodes of the LED's and puts these LED's in series. This will eliminate the problems with brightness differences in each of the banks. Another big benefit is that the LED's will draw much less current because they are in series instead if parallel. V+ for the LED Circuit must now be higher than the 3.3 or 5V Arduino Voltage.
Edit: You could also use N and P Channel MOSFET's which would reduce voltage loses and allow you to drive higher currents with much less heat dissipation than BJT Transistors will have.
