Generally speaking, for running you should never wear cotton. Once it gets wet it chafes and that hurts.
Wear nothing under your shorts. Most shorts have a liner that may be perfectly sufficient.
Wear short running tights. This is mostly for chafing to prevent the skin of the thighs rubbing together. You don't need to wear anything underneath. ...
Consider the following:
You should be non-allergic to the cloth material under two condition atleast (when it is dry and when it is wet/soaked with sweat)
Are you comfortable wearing it?
Does it make you feel good? (Because you'd be running long distances and this(inner-wear) should be last thing in your mind bothering you.)