Hi, beautiful horse! That guy in the picture looks like he's riding her mouth. His posture looks like he's riding a speed racker, and most people I know buy a Lane saddle for Saddlebreds-they are built differently.
I would spend the first few months just walking on a looser rein, and working on a gait faster than a walk with some contact. Contact meaning that you can feel her mouth with a soft rein, not a loose rein. Just practice going from the walk to the gait to the walk for a few months.
Just shortening the reins and some leg pressure should get her moving, and a tap tap tap with your heels should get her going. You can teach her cues that you want. I use one kiss to walk and 2 kisses to gait. If I want to go fast, I raise my hands a bit, and sit on my pockets a bit more. I also love voice commands. You can teach her to gait by saying gait.
The thing that took me longest to learn with my gaited horse is to do Nothing. I get her gaiting, keep my balance, and give her a rein contact to keep her balanced between my seat and hands. Then do nothing. Just not interfere with her gait. Let your seat bones slide back and forth with the gait.
I could look for balance forever. Then I read something that made a lot of sense and works really well. Your balance should be such that if your horse just "poof" evaporated from under you, you would hit the ground standing. Not fall on your back or fall on your knees.
One of the most important things is saddle fit. Find what kind of saddle you want (trail saddle?), then find a saddle that fits. My gf has a Rocky Mountain and she goes very well in a National Bridle Tennessean gaited saddle. My horse has a different build and goes nicely in it too. A poor fitting saddle will compromise her gait.
Hope this helps