new seeded lawn
I would selected a seed mixture with perennial ryegrass and fine fescue. Perennial ryegrass will thrive in the sun, while fine fescue will persist in the shadier areas.
Regarding fertilizer select a product high in nitrogen, low in phosphorus and moderate levels of potassium, such as 25-5-15.
May is a fantastic time to seed, and fertilize! Think about turning on you irrigation system in late May as well.