I, too, look for someone personable. I much prefer someone that I am comfortable with and can easily talk to. I once went to an instructor not knowing anything about her other than she taught basic jumping and western pleasure. Her students all hung in the horses' mouth and jerked them around. Ever since then, that's the first thing I look for, honestly. I look for happy healthy horses and a roomy arena.
I agree with Peyton, he has to be very personable! I love when they're with me and investigated with me even in shows. They also have to be horses lover :)
I would say the training's results he had, his experience and his capacity of making improving his students.
First off they must be very personable and look at me when I'm talking to them. Then I move on to watching a few training sessions to see if the horses they are working with look happy and the way I want my horse to look! And then make sure the facility is clean and the feed is stored properly