Totally second the idea of doing dressage exercises. But IME, riding a dressage test pattern is not that useful (unless you are trying to memorize it for a show next week). Also, there are a lot of useful and important gymnastics horses can do at lower levels, long before you need to “teach to the test.” Doing all the lateral at a walk (shoulder-in, spirals, travers, renvers, etc) on the volte and straight, for instance, is really good exercise for both of you from early days, even if you are nowhere near the test level that requires shoulder-in trot.
Also riding shoulder-fore, or just with a little inside rein and inside leg, can help keep a spooky horse on the rail. Try to be pro-active here, so that the horse can’t move off the rail when she spooks. This can help make the spooks smaller. Anticipate the “scary places” in the ring and ride through them in shoulder-fore.
This is a little counter-intutive, since I think if a horse spooks off the rail, our first impulse would be to use the outside rein to try to guide him back onto the rail. But this in fact lets the horse bulge to the inside, and fall into the inside even more. If you ride with a slight inside bend, on the other hand, and keep the haunches to the outside, the horse really can’t escape from you. It also keeps the horse from staring at the scary thing and working up a spook.
Spooks at familiar things can be learned behavior and a form of resistance (as opposed to spooks at truly scary things in strange environments). I would suggest not taking them for granted as a quirk of the horse, but rather figuring out how to minimize them. Your instructor should be able to help here, too.
Riding the “scary corners” in shoulder-fore is also a way of giving her something to think about.
As far as trails: she might or might not be spooky on trails. Try to go out in company with other people a number of times. Ride behind another horse. Both of you will just relax. Also, why not go out on the trails for a handwalk or a grass walk? Get to know the area from the ground, and you will both be more comfortable in the saddle.
All that said, it can be hard for a beginner rider to know what to do in a schooling ride, because you can’t really see the smaller things that need working on.
I really think trails are your friend here, if you can work your way up to them without having a big scare. You can school half an hour, then do a trail ride and work on your own position, posture, leg, etc. while the horse happily moves forward.
Choose days for a trail ride when the horse has proven herself in the ring to be happy and attentive, not days when the wind is blowing and she is spooking at her own shadow 
It is so easy these days to get stuck in the ring, and to be afraid to ride outside it. Even to get stuck in the indoor ring. I see this happen to alot of adult riders, and it is too bad. Work on it; set up trail riding as another kind of “lesson” for yourself, even if that means riding with a barn buddy, not an instructor. Approach if in steps. Etc.
