Like Koffee and the Major said, speed is important and actually helps you balance and correct. Your instinct is to stop or pedal slower when you start too feel off balance, but DON'T...pedal faster.
Everyone falls learning, just like you did learning to ride in the first place. I actually learned on handmade wood rollers with wood drums that must have been 8" diameter. I learned on a touring bike with a rear Blackburn rack, and had a friend that would stand beside me and steady me using the rear rack (still fell a few times).
Hang in there...once you learn you'll never go back to a trainier!