For your stepper motor, the black and green wires make up the ends of one motor coil, and the red and blue wires make up the ends of the other motor coil, so one way to connect it would be:
1A and 1B connect to black and green (which goes to which does not matter)
2A and 2B connect to red and blue
As far as making connections to your Arduino, the simplest scenario requires two I/O lines from the Arduino (you can choose any lines you want), one connected to the motor driver's STEP pin and the other connected to the DIR pin. Since your Arduino has a built-in regulator and is likely being powered separately, you do not need to connect 5 V (VDD) from the motor driver board to the Arduino, but you do need to make sure they have a common ground (i.e. connect GND from your Arduino to one of the GND pins on the motor driver board). Other than that, you should short the VDD pin to the 5V pin above it as shown in the motor driver's simple connection diagram:
This powers the board's logic. You will also need to connect a motor power source to VMOT and GND. Does this make sense?
You might also find this thread helpful: https://forum.pololu.com/t/a4983-and-a4988-getting-started-guide/2789/1
Please let me know if you still have questions.