I had to look at some data for a similarly rated relay recently. Now this might not be gospel, but 70mA to energize the relay I had to replace was more than sufficient.So to answer your question, just using the Arduino related wallwart would be enough.You say you're new to electronics, so I hope you understand that the switching terminals on the relay need to be wired in the correct sense that when no power is applied to the relay, the grid input is switched off too.UPDATE:forgot to mention that you should add some decoupling to the relay coil to reduce switching spikes
Measure the resistance of the relay coil with a multimeter, then use Ohms Law to calculate the current it takes in the steady state @ 12V. The initial current when the relay turns on may be a little higher because of the movement of the relay armature.You should be able to use a single power supply for both the relay and the Arduino.
I believe I need the relay to have the grid input working even when there is no power to relay, this way if my arudino fails I can keep the grid input working. I understand what you're doing here. Most grid converters are intelligent anyway. I forgot about that.Also helps me to override the arduino incase something goes wrong because I do have a seperate manual switch before the line goes to relay, just in case. Yep, good.Im switching off the grid input just to disable the Inverter's charging function during the day and it should use the Grid when there is no -sun. I do not want to drain my batteries unless required.This is more of an UPS than an inverter.Let me know if im wrong here. Nothing wrong about the above.I have only added a Diode to the relay coil and is this enough? Yep, does the job. Sounds like you've researched this well