How can we turn hard dirt into a green oasis?

you have not mentioned a location?

If your soil is very hard you may want to rent a rototiller and loosen it up. You can also use it to help mix in soil amendments and fertilizer... you will need to do this whether you decide to seed or install sod.... sod can give you an instant lawn... without having to keep the kids and dogs off of it while the young fragile grass gets going... and there is less chance of weeds.... check with a LOCAL nursery to get the correct amendments for your soil (take in a sample) and information on which grass type does the best in YOUR area...they will also give you the instructions on planting or seeding, installing, and watering and maintaining.

https://www.youtube.com/watch?v=T1BdRowei20

The entire yard needs to be tilled but very carefully since you have a sprinkler system in place. You need to get this done asap, as AZ will be very hot soon and the seeds could fry, not germinate. Here are some great step by step sites.

https://yardcare.toro.com/create/planting-a-new-lawn/starting-from-scratch/

https://www.thespruce.com/how-to-start-lawns-from-seed-2132204

http://www.dummies.com/home-garden/lawn-care/planting-a-new-lawn-from-seed/

Also contact your local Ag Extension, they will have excellent local information for you too - specifically type of grass seed to use.

Hi Debby this might be a pain but it works I have the same problem. Go to your HD, Lowe’s, or Walmart buy bagged top soil & peat moss. Turn over your lawn about 2-3 inches down ( to remove weed roots). Mix your dirts w/ top soil & peat moss. Though peat moss beads water away once it starts to break down it softens the dirts. Usually hard pan dirt is caused by salt from the road & a lot of rock in the soil. When I did mine I actually screened my dirt to remove all the rocks. & where I did it the lawn came in nice & rich. But remember you have to maintain it cause new grass burns. Never water during the day that also causes the roots to burn. Weed feed fertilizer @ least spring & fall & if your Walmart carries a product called worm poop get it. It’s the best fertilizer. You attach the bottle to your hose. I do it once a week. & pp, ask all the time who is my landscaper. Lol

Rototill topsoil, compost, and peat moss into your dirt.

I agree with William, till with cow compost, about 3" compost. Till deeply, then till again the opposite direction. THEN buy earthworms and plant them in areas around your yard. Just a couple worms and some fruit or vegetable matter in per hole a few inches deep. Continue to water your soil as if it had grass. It will soften

Hard to answer not knowing your location. But if grass is the future begin with tilling up the hard areas you want for grass and then plant seed and water. for areas you want to plant establish a border to separate the grass from this area and begin to feed that soil prior to planting.

My bad just noticed you're in Arizona - what a challenge thinking a neat native garden would be fun in conjunction with native grass... In Texas Bermuda does well in hot climates.