How fast grass seed sprouts depends on how warm the soil is and whether you keep it moist at the surface, whcih requires watering twice a day. Under ideal condiitons, you should see a nice haze of bright green baby grass coverin the ground in 2-3 weeks (assuming you use fresh seed labeled for a high germination rate).
For the lawn to be established enough that you don't have to worry about keeping it moist in warm weather and that is tough enough to walk on, allow about 3 months.