I'm currently pursuing a BS in physical therapy, focusing on training healthy individuals. I think we're kinda on the same page concerning where we want to be after we finish our degree so my answers will reflect what I want for myself in terms of training healthy people.
Able to identify function/dysfunction in all movements. Have a good understanding of all aspects of fitness and health, including strength training, aerobics, shock training etc.
Have at least a mediocre understanding of nutrition, because you can't prescribe exercise without prescribing a nutritional program to match. Unless you like banging your head against a wall.
In depth knowledge of anatomy and muscle function, of course.
Understanding the difference in needs of the never-moved-my-ass-in-my-whole-life crowd and elite athletes, and being able to handle both.
Perfect technique in all big compund lifts, and understanding of when to use each one. Knowing the difference between free weights and fixed plain equipment. Know when to use which.
Know which other health professionals they should work with. Including massage therapists, art therapists, doctors and physical therapists.
Personal trainers, school gym teachers, athletes, weekend warriors. Just basically all those who instruct other people in training or train themselves.