Why not both at the same time? SpaceX is clearly set on going to Mars. There are other organizations interested in doing something on the moon. There is really no need for all these organizations to pool their resources and focus on one thing.
The psychological and medical issues are already being researched on earth and on the ISS as well. What difference would a moon base make in that respect?
Sending ships from the moon would not be a lot better actually. It would be a waste of time and fuel to land ships on the moon, refuel and relaunch them from the moon. You can however, possibly, manufacture fuel on the moon and bring that fuel up to an Earth Moon Lagrange and refuel your ship.
But it is not necessary to do that in order to get to Mars so that certainly isn't a reason to wait. Plus it would probably take a long time to set up facilities.
There is very little benefit to establishing a base on the moon first. Any thing that can be learned on the moon that can be applied to a trip to and a stay on Mars can be learned from experiments on earth and the ISS as well.
Did you read or listen to anything from Dr. Robert Zubrin? There was a video link posted earlier. I think the part starting here especially addresses some of your points
https://youtu.be/EKQSijn9FBs?t=37m39s