so I just planted new grass and it's been about a month so the grass is doing well. I'd like to switch from my twice a day for 20 minutes on each sprinkler to a two times a week-ish and longer, 30 minutes or so, to encourage deeper roots.
Any experience on how long the sprinklers need to go to properly encourage deep roots without wasting water?
Length of time isn't what you're looking for, it's the quantity of water you apply. Ideally you want to apply about 1/2" of water twice a week. Really though it won't matter a lot this fall because the new roots won't get very deep until next spring anyway. Be sure to put on a good winter fertilizer look for help encourage root growth.
With brand new grass, keep the soil moist until the seedling grass establishes. Also, use a fertilizer with high ammounts of phosphorus and potassium. Some areas ban phosphorus, but most have exceptions for establishment. The fertilizer breakdown is nitrogen, phosphorus, potassium. A good example of starter fertilizer is 20-20-10. After establishment, switch the watering to 1/2-1" of water, once or twice per week, early mornings.