I have everyone beat now that I'm past 60.
Been riding all my life but only got halfway good at it over the past half dozen years. And by the way, OP, that does NOT mean it'll take you that long. The best way to progress fast is to find a really good instructor and ride as much as you possibly can. If you're like me, the more you ride the more you'll want to ride! Also, I gotta say it, if you can somehow get your own horse, nothing compares to that. When I didn't have my own horse, no matter how easy it was to get to the barn, I always felt a little shy about it, having to ask if so-and-so was available to ride, etc.
But if you're not at that point yet, I second the combination of yoga and Pilates. Balance and core strengthening are what you need. Besides those, lunge lessons are your friend. Any way you can start doing that? I didn't see whether you were at a large barn with other students; if you are, maybe you and another student could trade off lunging each other. And does your barn allow practice rides at a reduced price? If so, really take advantage of that.
Personally I found bike riding wasn't helpful for my equestrianism, it required different balance and used entirely different muscles.
Finally, the one book I would recommend to anybody who really wants to get their head into how to be a great rider is Sally Swift's Centered Riding.
She also has DVDs and has certified instructors who teach her system, and you can learn a LOT from both. My one Centered Riding clinic made a permanent difference in my riding.
Good luck and many happy years of riding to come!