Plant recommendation for underneath dogwood
You have a nice-sized dogwood. Anything you plant under it will have to compete with the dogwood's roots for water and nutrients. The rule of thumb, anywhere, is to keep a plant watered for at least the first 2 years when rainfall is not sufficient. Plants need about a inch of water a week. Soil can dry on the surface, but be moist below--about as moist as a wrung out sponge.
All plants below the dogwood should tolerate shade. We'd recommend low shrubs and groundcovers. Natives are good. Avoid invasive plants like English ivy or highly competative ones like pachysandra. Epimedium, ferns (native), phlox stolonifera, and iris reticulata.
The online publication "Native Plants for Wildlife Habitat and Conservation Landscaping" is a great source with photos and cultural requirements for each plant.