Question
How do I get grass to grow in my yard with dry, clay soil and some trees that shade it?

Answer
Whether you choose seed, sprig or sod, that clay soil will require some tilling. Seed will require that you prepare a seed bed - the soil particles must be quite small to a depth of at least three inches (six is better). Sod will require the less refinement; the soil particles can be larger. Sprigs lie somewhere in between. Then there must be irrigation. With seed the watering is critical. The seeds are all sown at approximately the same depth and if the watering fails at that depth, there will be almost total loss. Myself, I would choose sodding. Clay soils do not allow easy water movement downward so the soil layer just below the sod will stay wet longer. As the turf becomes rooted, irrigation management will be the key to success. When clays get wet, they swell and as they dry out they shrink. This permits water movement downward and root development. Too much drying out can cause the young roots to break as the clay shrinks. But this swelling and shrinking, together with root proliferation of grass crops causes the improvement of clay soils over the long (very long)term. Addition of organic matter during the preparation stages will be of immense help.