The " cause of action" is the reason you go to law. It covers the wrong you allege you have suffered and the remedy you hope the court will apply.
"Legal proceeding" is a formal step you take intended to lead to a judicial decision.
"Legal action" is more or less the same as "legal proceeding", but is perhaps wider and could include some actions that do not require judicial intervention, e.g. forfeiting a lease or levying distress for rent.