Seed first and let germinate and establish first if you can. How long to wait after will depend on whether the product contains a post-emergent or pre-emergent herbicide. With a pre-emergent you will have to wait until fall at least.
And remember fast germination is not always what you want. ANNUAL rye will germinate in around 5 days but will only grow one season Hybrid PERENNIAL rye will appear in 7-10 days. Bluegrasses and fescues can take 14-21.
Now is a good time t buy seed because you can get fresh seed in the custom blend you want in bulk and cheap.