Need help finding out whether this my circuit will work as intended (new to electronics)
There are two obvious problems.

1) The most important - you need a signal ground connection which will connect to the Arduino ground. If you are using a 5 volt supply and sharing it with the Arduino you're all set, but there are many different ways to connect to the many different Arduino models, so this is not guaranteed. Add a signal ground and use it.

2) The resistor common should be connected to V+, not V-, and the LED polarities should be reversed. That way, when an output is pulled low and current is flowing through the winding, the corresponding LED will be lit.