Considering the small difference in cost between a test kit and the Extension Service's test, be smart, get it done by the professionals and have a printed report you can use to make decisions. Neither clay soil nor alkaline soil can be successfully changed into another type for more than a short time. The corrections must be applied over and over for the life of the tree and the corrections are a stopgap at best. Don't do that to yourself. It's too much work.
Most shade trees fall into the canopy category and require full sun to do well.
Understory trees can take light to half shade. Be realistic about the sun in your yard. Choose a tree that is adapted to that level of light and it will thrive if the other two conditions are met. In our area of Minnesota, the average annual rainfall is 30 inches. If that is all you plan on the tree getting, choose a tree known for drought tolerance, but remember that any newly planted plant will need to be kept watered at the level of about one inch weekly during the first season while it is establishing new roots in your soil.