You can do one-legged drills on your spinning bike, or your trek up on a trainer, to help you master your pedal stroke.

In terms of tires, I put cyclocross tires on my hybrid and love that it gives me the option to ride on the trails when the weather prohibits a road ride. If you decide to do this, you might actually want to find an "easy" trail (unpaved, but not very technical) to practice on. You (probably) won't have as many issues with black ice as you will on the road - and practicing off-road will help you master your bike handling skills as well.