The basic process in preparing to sod is to till the soil, then rake it smooth (removing any rocks, loose roots, or other debris on the soil's surface). Then lay out your sod. Keep it well watered (daily, if the weather is warm and dry) until the sod becomes established. This generally takes a minimum of a few weeks.
After 2, 3 weeks, fertilize the lawn to help it become established more quickly. You may find it helpful to wet the soil just before you lay the sod. Avoid making it muddy; just water it enough to keep the soil from being dust-dry. The moisture makes it less stressful for the sod. Concerning the tree roots, it would be hard on the tree (and your back!) to remove the roots; instead, build up a thin layer of soil over the roots. You want to use just enough soil to create level ground for the sod. Be careful to avoid covering the tree roots with more than a couple of inches of soil, or the roots will suffocate and the tree might die.
Avoid rototilling the soil within the root zone of trees. Mulberry trees (and many other kinds) are tough on lawns. Any sod you lay right under the tree will probably fail to thrive. The shade compounds the problem. Instead of grass, consider mulching around the tree or adding a shallow-rooted ground cover.