This current limiting resistor should be > 220 Ohm in order to protect the Arduino output (< 20mA). It also should be low enough to feed the transistor base with enough current for the LEDs. What's the LED current?
In that case, would it not be better to have some sort of diode in place of the resistor? Iām not too sure about the LED current yet because it will be a bunch of garden lights that are TBD. Should that be a consideration? I thought that only mattered to spec the appropriate power supply?
You need a resistor in order to limit the current flowing out of the Arduino pin to not damage it.
Let's say the max current allowed is 20mA. The transistor you show has a max "beta" (current gain) of 40. I'll make the assumption that it will be at least 30 in this application. Meaning that with 20mA flowing into the transistor base, you will get 600mA (20mA * 30) through the transistor for LED current.
Assuming 600mA is enough for your LEDs, then to limit the current into the transistor base to 20mA, the resistor will have 4.4V across it (5V out of the arduino and 0.6V at the base) so 4.4/0.02 = 220 ohms.