The resistor in the circuit you referenced is needed to limit current to the relay input, else you could damage the Arduino.
Not really understanding your second question about connecting the grounds. All wires that end with the ground symbol should be connected together: the negative terminal of the battery, the Arduino GND, and the relay CN3 terminal.
You probably don't need the resistor but it probably won't hurt either. Most of these solid state relays, such as the one in reply #8, accept a wide range of input voltages. They are not rated for a specific current, so a current limiting resistor is not needed to protect the relay. It is not likely that the relay will draw enough current to damage your Arduino.
I supose the CN3 and the arduino goes to the ground from those not connected cables... but where do I connect the ground?
The 'ground' symbol just signifies a common connection point. Connect all of the wires that go to a ground symbol together. You do not have to connect the ground to anything else.
Also, to use Flash you are going to have to install some version of serial proxy like tinkerproxy or such. There are several flash libraries available to communicate with the arduino. My advice is search for arduino and flash. More than likely you will find Mike Chambers recent tutorials - they are an excellent place to start. He is also on these forums as well.
