You need to connect the -ve of the 12 V supply to the 0 V of the Arduino.
To make the circuit reverse the motor, you need to duplicate the relay, diode, resistor and transistor. The ground and +12 V of the second set are connected to the same places as the first set. The Arduino end of the resistor needs to go to a different pin of the Arduino so that it can be controlled separately.
Connect 12 V to pin 87 on both relays.
Connect 0 V to pin 87a on both relays.
Connect one end of the motor to pin 30 on one relay and the other end of the motor to pin 30 on the other relay.
Activate one output on the Arduino to make the motor go in one direction. Activate the other output to make it go in the other direction. If neither or both outputs are active, the motor will stop.
If you are interested, the transistor, the TIP122 is acting as a switch. When turned on, current flows from the 12 V supply, through the relay coil, (pins 85 & 86) through the transistor to ground. When turned off the current stops.
To make the TIP122 turn on, current has to be fed into the "base", which is the control pin on a transistor, and is the one that the resistor is connected to. You only need a current about 100 times smaller than the relay coil current to be fed into the base, so the Arduino provides that smaller current.
The voltage on the base of the TIP122 will only be around 1.5 V, so the resistor is there to limit the current and is more to protect the Arduino than the transistor. (The TIP122 is actually 2 transistors in one package, called a Darlington Pair. The base voltage is around twice what it would be for a single transistor, but the need to limit the current is much the same).
The diode is a "freewheel diode". The current in a coil, such as the relay coil, can't change very quickly. When the Arduino output turns off, the TIP122 turns off very fast, and that will try to stop the current flowing in the relay coil. That could cause a voltage surge, which could damage the TIP122. The diode lets the current continue to flow in the relay coil, removing the voltage surge. The current will stop more gradually, over maybe 50 ms rather than less than one microsecond.
The current in a coil behaves like it has inertia. If you think of it like a ball rolling, turning off the transistor is like putting a wall in the way, there is an impact. Having the diode is more like no longer pushing the ball along, and letting it run to a halt in its own time.
There is one disadvantage to a diode. It slows down how fast the current reduces in the coil, and it can cause the contact to open slowly in the relay, and the contacts get damaged more than if they open quickly. If you replace the diode with a resistor of 470 Ohms (or two 1 k Ohm resistors in parallel) the current will stop faster than if you have a diode, but not so fast that there is a big voltage surge.
Think of it as the ball rolling into syrup. It's not an impact like the wall, but the syrup stops the ball faster than it would on a smooth surface.