74HC595 will not work well for this, the output current it can support is too low and the LEDs will be too dim.
Find common anode RGB LEDs and use TPIC6B595 instead.
Or use parts with more outputs, like WS2803, 18 I/O, can drive 6 RGB LEDs each.
tpic6b595 open drain shift register.pdf (150 KB)
WS2803-preliminary-En.pdf (437 KB)