If having fallen off means you are an expert, I was an expert by the age of 12. My pony probably dumped me 50 times the first year I had her. I was a stubborn kid, and I just got back on. I am now much harder to dump..
I also will not get on a horse that is acting like an idiot. I have no desire to break things. My horse may spook or bolt a few steps, but she pulls in easily. The one spill that I took off of her surprised her more than it did me.
Over the (many) years that I have been riding, I have taken dressage lessons, jumping lessons, western lessons, and done a lot of trail riding. All I am really interested in is trail riding. On a trail, anything can happen to scare a horse. But trail riding is what I love.
I would suggest not just sticking strictly to lessons, but get out and enjoy a nice ride in nature. That is my definition of riding.