Cute mare. She has nice conformation and could probably do the lower levels of hunters, eventing or jumpers with the right training.
However, PLEASE find a trainer. You have a very dangerous setup there; with the curb bit, lawn chairs and no instruction. I was there years ago--a lifelong Western rider with no jumping instruction, a bareback pad with stirrups (*gasp* the horror!
) and a fantastic, compliant horse. I can't tell you how lucky I was we were never injured, and to this day I wish I had gotten proper instruction earlier.
As far as what to work on when you find an instructor, you'll most likely be doing a lot of grid work, starting with poles on the ground and moving up to cross rails, working on lengthening and shortening your horse's stride, and doing a lot of two-point at the trot without stirrups. This is something you can practice on your own to build strength in between lessons. It is HARD work, and your thighs will double in size (okay, maybe not THAT much...), but it will help you tremendously.
A word about the jumps--especially in the early stages of learning, it's really important that fences are safe and fall easily to prevent an accident. The lawn chairs are unsafe because, not only will they not allow the rails to fall, but a horse can catch a hoof between the holes if he misjudges the fence.