Try a LED connected to an Arduino pin with something like a 560 Ohm resistor and see how it looks. You can user lower values for more brightness, but without knowing the exact specs of the LEDs I wouldn't go below 220 Ohms.
6 LEDs at 15 mA is only 120 mA so your power supply is probably fine.
I don't think you'd need to control the LEDs with transistors if they're generic 3mm or 5mm ones; if you have enough Arduino pins free then you should be fine driving them directly. You'd only need transistors to switch higher-current ones, or a bank of LEDs.
Good luck!