For example, handling a water bottle while riding (see the blog, "Where's Gunga Din When You Need Him?", 04/01/16 -- and that wasn't April Foolin'). That can be especially tricky in a strong side wind.
Something I think would be good to learn is, getting things out of a jersey pocket while riding. Reaching around behind me to dig a snack out of a pocket would be much trickier, I think, than riding one-handed to take something out of my top-tube pouch.
I have several problems with that. One, on a ride long enough for me to need to carry snacks, I'm using the 100-oz Camel-Bak. It's so big that it covers the jersey's pockets. Two, my jersey pockets aren't so very big. I'm small for an adult, and smaller jerseys of course have smaller pockets. Or maybe pocket size varies according to brand, and I need to try a different brand.
Fixing a flat? I'm not one to talk. I barely know how to change a tube, and have had very little practice at it. I use tire liners to reduce the risk of punctures, and am reluctant to experiment on my bikes. I'm afraid I'll wreak havoc.
There are books out there, such as Complete Bike Maintenance, for the mechanically inclined. I think I should learn more about how to tinker my own bikes, but as already stated, I don't want to ruin anything. If I got a windfall that let me afford a new road bike, the old one could become my guinea pig. But is a three-year-old bike OLD?? Gimme a break. It hasn't got 10,000 miles on it yet!