Dogwood trees require a fertilizer that adds acidity to the soil. This provides the best conditions for a Dogwood to grow. Look for fertilizers that contain forms of ammonia, phosphoric acid, or urea.
You can use any regular fertilizer just make sure you dont overdo it. Too much fertilizer will not make them grow any faster and will actually kill them instead. Use 1 tablespoon full in march and then maybe again in July. Also make sure the trees are draining properly as too much water can also kill the trees.
Pink dogwood trees are very easy to care for and require little fertilizer. use less then a cup of 12-4-8 fertilizer in the first growing season. overfertilizing may stunt the growth of your tree.
It is not recommended to fertilize dogwoods the first year they are in the ground. Try a low-nitrogen fertilizer next spring and then every other year.
A balanced fertilizer should be applied in the spring before it blooms, or late fall. If the tree is small I would put some kind of protection around the tree so it doesn't get damaged from ice or snow.