Leo, RI You need to top dress at every opportunity with the eventual final mix of grasses. I suspect that you may have rye grass, if it came up so quickly. A good mix for New England is blue grass and fescue in addition to perennial rye. Perennial is a tough grass for high traffic areas. Blue grass will do better in sunny locations but requires more attention.
Fescue is a fine grass that grows in poor soil and in dry conditions.
A mix that accounts for these conditions will survive. Top dressing with loam and new seed will crowd out the weed seed, which is the bane of a good lawn. One dose of hydrated each spring will suffice in sunny areas. Those four step programs are not necessary, if you can keep the weeds from getting established and keep the grubs away. A good test for chinch bugs and grubs is to put a coffee can, with the top and bottom cut out, into the ground.