Hook-up the solenoids up to the Arduino as in the schematic: connect the TIP120s to digital ports 5 and 6 with 2k resistors in between. One pin goes to ground, the other goes to the solenoid, with a diode across it (make sure you get the polarity right), and to the 12v supply in the end. The other wire of the power supply is ground. Connect that to the other grounds. Just connect all grounds of all components including the Arduino together.
Over here you can download code to specifically test the two solenoids.