There are two options. You could connect them to individual Arduino pins, or you can connect them through the transistor. The former gives you individual (or paired) control. The later avoids any current draw issues from the Arduino's microprocessor.
For the most part, the Arduino's 5v power supply is strong enough to power each of those leds at 20mA without much issue. That's 280mA. But given that typical red led's have a forward voltage drop of 1.8~2.2V, they are assumed to be 2v leds. That means you can pair two of them in series, to share the current. Instead of 20mA x 14, you could have 20mA x 7. More energy efficient that way.
The 40Ω resistor is based on R = V/I, where V is (Source Voltage [5v] - Led Forward Voltage Drop [2v * 2 because two in series] - Transistor VCE Drop [Typically 0.2v in saturation]) and I is 20mA, so (5 - 2 - 2 - 0.2 = 0.8) / 0.02 = 40Ω. Round to standard resistor values, 39Ω or 47Ω.
The same can be done with direct connections to the Arduino gpio. 2 leds in series with one resistor per gpio. Reduce the current to 15mA each (68Ω) and you are well within the Arduino's ATMega's limits.
simulate this circuit – Schematic created using CircuitLab
As for the TIP120, you can use that, but as the saying goes, that's like using a bazooka to kill a fly. The TIP120 is a darlington pair transistor, and is suitable for up to 5 Amps. A tiny 2n3904 can handle 100~150mA (200mA at the highest), while a 2n2222 is better at 1A. Both are fairly common and dirt cheap.
In either case (14 individuals at 280mA or 7 strings at 140mA or less), the amount of current does not require the use of a TIP120.