Probably 90% to 95% of the lawns in the greater Phoenix area are one form of bermuda or another. New homes start with TIF 419 type of hybrid bermuda to have an instant lawn. 419 only comes as sod and should cost about $50 per pallet. Seed is much cheaper but the grass becomes shaggy with seed heads. 419 has no seed heads. Then as trees grow and people neglect, other forms of bermuda may move in or out. If you have 6-8 hours of unfiltered, direct sunlight, then you can grow bermuda just fine. If you have sissoo or palo verde trees in the yard or over the south, east, or west fences, then you will have to go with St Augustine. St Augustine is not the water hog people will tell you it is. If you want to have a green lawn, bermuda or any other, you will have to give it an inch of water a week most of the year and 2 inches when the temps are closing in on 110 or more.

I'm not certain gardengal is familiar with your soil. Six inches is a luxury especially starting with what you have.

Another possibility for full sun grass is UC Verde buffalo grass. That also only comes as sod. Don't know if it is available to you there, but it would be acceptable in full sun.

For right now, I would mow the weeds with a mower and string trimmer around the edges. Start taking care of it like it was a lawn and it will look much better. Furthermore once you start mowing and watering, some of the weeds will actually disappear. Some weeds, especially sand spur stickers, cannot take any care at all before they die. The roots of those weeds are very important in improving your soil. With temps in the 80s you can water it every other week. Coming very soon will be temps in the 90s. Then go to once a week watering, 1-inch, all at one time. My favorite sprinkler is an oscillator on the end of a 3/4-inch hose. Get the "turbo" type oscillator that does not have the exposed mechanical linkage. Those linkages can break or jam at any time...even the very first time you use it. The turbos will go for years without messing up.

Watering: Deep and infrequent is the mantra for watering. This is for all turf grass all over the place. Deep means 1 inch all at one time. Put some cat food or tuna cans around the yard, and time how long it takes your sprinkler(s) to fill all the cans. Memorize that time. That will be the time you water from now on. My hose, sprinkler and water pressure takes 8 full hours to fill the cans. Your time will likely be less. I like gentle watering. As for watering frequency, that depends on the daytime air temperature. With temps in the 90s, deep water once per week. With temps in the 80s, deep water once every 2 weeks. With temps in the 70s, deep water once every 3 weeks. With temps below 70, deep water once a month. Note that you have to keep up with quickly changing temps in the spring and fall. This deep and infrequent schedule works in Phoenix and in Vermont, so it should work for you. The reason for deep and infrequent is to grow deeper, more drought resistant roots and to allow the soil to dry completely at the surface for several days before watering again. If it rains, reset your calendar to account for the rainfall.

Mowing: Mow bermuda at the lowest setting. May as well mow your weeds low, too. Mow every week.

Fertilizer: Wait to fertilize your weeds until the end of this month. Phoenix is way ahead of the rest of the country on timing things. The fertilizer you use is less important than most anything else, especially if you are growing weeds (for now). If you want to use an organic fertilizer, that would be great.

Just out of curiosity what part of the area are you in? I know it sprawls everywhere. Oh, and do you have that shade thing with trees around the yard?