Question
I want to root some cherry trees from and existing tree. When is the best time to begin rooting the cuttings?

Answer
The best time would be early spring.  Cherry is one of the hard trees to root. If it was me I would try. Use new growth and cut these about 6 to 8 inches long dip them at once in a rooting hormone  (indolebutyric acid (IBA) would be best. Check with your local garden/ nursery type store for this.
Here is a web link that gives details on a method of rooting cherry trees.
http://www.gardenguides.com/101369-root-cutting-blossoming-cherry-tree.html