One way to improve your balance and core strength is to get a stability ball (aka yoga ball, exercise ball, etc). An exercise on it that is very similar to riding a bike no handed is to balance kneeling on the ball - but don't expect to start there. At first just practice sitting on the ball and try the exercises that are usually packed with the ball (or find some online).

When you feel ready try balancing on the ball in stages. The first step is to be able to balance on all 4's on the ball. When you can do that work on holding one arm at a time out for a few seconds. When you can do that work on holding one leg out at a time for a few seconds. When that is comfortable try holding out the opposite arm and leg at the same time for a few seconds. If you can do that try sitting up kneeling on the ball (shins on the ball, thighs extended, butt off your calves). Work your way up to being able to kneel on the ball for up to 2 minutes. You probably will fall off the ball at some point in time, so practice this in a safe place.

My coach started me on ball exercises a few years ago for core strength. I got my husband to start doing it too and not only did his occasional back issues clear up, he's pretty sure that the balance exercises have saved him in races a few times, when he's been bumped hard.