Start watering it soon. Put out some cat food or tuna cans and turn on your watering system (sprinkler or in-ground). Watch for the cans to fill up and time how long it takes. That is your baseline for watering. Water it that long once per week. Then mow every week. Watch for seed heads to form on that coarse grass and take some close up pictures. Get the camera about 3 inches from the grass. Seed heads make it much easier to identify a grass.

The time to renovate would be in late August for you in NH. It is also the time to overseed with fescue seed if you have that. The color leads be to think you have Kentucky bluegrass.

Grassy weeds, like what you seem to have, are hard to get rid of in a grass lawn. It usually requires renovation where you kill the entire existing lawn and put down new seed.