What is the best kind of shade tree for a backyard in growing region 6?
There are many great trees homeowners in growing region 6 may want to consider for their backyard depending on their priorities. 

First, the two main types of trees are deciduous, trees that lose their leaves each winter, and evergreen, which retain their foliage year round. Many of the most popular shade trees are deciduous trees, however there are some great evergreen options as well.

Starting with deciduous trees, there are flowering trees and traditional shade trees. Of the traditional shade trees, you may want to consider maples and oaks, of which there are many varieties. Red maples grow a bit slower than silver maples, but have hardier bark, beautiful red leaves in the fall, and a more pleasing traditional branching habit that many prefer vs, Silver maples. Sugar maples growth slower still, but are beautiful, prized trees that turn glowing orange in the fall and produce the sap used in maple syrup come spring. These trees can be 40-70 feet tall at maturity. Oaks are incredibly beautiful trees that are typically a little slower growing than maples, but have strong bark, enabling them to survive harsher conditions and have a long lifespan. They can handle being used as street trees as they do fine with moderate pollution levels and salt from de-icing. You really can't go wrong with maples or oaks as a summer shade tree, with their tall canopies providing wonderful shade once their leaves fill in during the spring.

Some popular flowering options that provide ample shade include cherry and magnolia trees, both of which have beautiful showy spring flowers and reach a height of 20-60 feet at maturity depending on which of the many variants you choose (be careful with this as some are best suited to warmer climates and don't tolerate the cold as well). 

For evergreens, the options are a bit more limited as many of these trees have low canopies that extend all the way to the ground. Many species of spruce and fir tree can have their canopies raised as they mature, creating space underneath for shade. Some pine trees have ample canopy under them, however their thin needle density can limit the amount of shade they provide.

Whichever you ultimately choose, Shade trees add a ton of value to a home and are a great investment. Consider working with an arborist to make the decision based on your lighting and soil conditions. Enlisting a professional's help with planting will ensure your new tree is set up to grow healthy and strong.