Old-fashioned gardens are full of plants that our grandmothers grew because they enjoyed the plants, native or not, that were reliably hardy and easy to grow. They were also often available, not at the local garden center but as "slips" traded by neighbors. Dogwood tree suckers fall into this latter group. Dogwoods are attractive trees that are readily available and---when they are native varieties---are easy to grow. To make it even better, dogwoods accommodate by producing numerous suckers each year.
Choose a sucker that grows out a bit from the tree to minimize damage to the parent. Most dogwoods will set out suckers that are easily controlled by mowing---choose one of these and let it live. Cut or mow other suckers as usual.
Sever the sucker from the parent in the fall by plunging a sharp spade in the ground until you can feel the umbilical root (called a "stolon") break. Do not dig the sucker up; your purpose is to simply separate the child from the parent so it can get started on its own root system. Tap the ground back in place.
In spring, carefully dig around the sucker, taking as many roots as possible as you lift it. Fill the hole with garden soil and water it well.
Dig a hole larger than the root ball for your new dogwood tree. Locate it in its permanent position if possible. Line the hole with a half-and-half mixture of garden soil and compost, well-rotted manure or peat moss.
Set the sucker in the hole and bury it with the amended soil so that it sits at the same level as it did near its parent. Water it in well and trim the sucker back to half its height to minimize shock and encourage branching.