We moved into our home about a year ago, and the front lawn was a mess. Mostly crabgrass and weeds. Because of this, we decided not to waste water on it during the summer, in anticipation of getting a new lawn eventually. It's a small lawn, less than 200 sq. ft.

The summer heat turned it into a mostly dry, yellow straw patch. We've had a few rain showers in the past week, so I'm thinking now is the time to get some new sod. Am I right? And I have some questions:

1. Tilling - I've heard conflicting things about tilling. Should we till or not? I think it would be great if we didn't have to! It's so much work. But we have clay soil, so I don't know if it would be prudent to till to allow the new grass roots to break into the soil more easily. But I've heard tilling encourages weed growth also.

Could we just cut the existing grass down to a stubble and put mulch over it and sod over that?

2. What kind of sod is best for the California central valley? It would be nice to have something drought resistant since we're so hot/dry here in the summer. And I'm also concerned about the crabgrass coming back, so something dense that would crowd it out, maybe? any suggestions?

thanks!

Sod will not grow on top of old grass. That's the first thing you have to get rid of. Grass has to put down roots and it cant if it has to drill itself through hard clay soil holding old sod. Also, the clay soil as described, sounds like you have cement where grass should be.
Tilling is the only thing you can do but further to that you have to ...you must....incorporate organic matter to make the grass want to grow and hold on to moisture. Organic matter comes in many forms; i.e. peat moss, newspaper and compost. Without going too deep into this you should do research on it.
Without purchasing expensive compost, consult your soil dealer about what is called triple mix...a combo of different soils including sand.
Whenever you see the word "SAND" used in conjunction with gardening, always think 'coarse sand', not beach sand, or playbox sand...that holds too much moisture. Coarse sand is larger and allows for good drainage. Coarse sand can be bought at soil dealers or from quarries--the place where gravel is bought.
Drainage is one of the most used terms for good turf growth. Water should run away from the house.
So it sounds like you have to till and one of the easiest ways to do that is to rent a roto-tiller..the size is dependent on what kind of soil it has to bite through.
Into the soil which you dig down at least 6"...or more...you inject the organic matter. If you dig down 6", put at least 2" of organic matter in and turn it over. Then look to levelling and seeing to good drainage.
Weeds, if you see them, pick out and garbage them.
If your lawn is mostly weeds right now, I suggest you kill everything green there by the use of a glyphosate..i.e. RoundUp herbicide...it will kill the weeds, the grass presently there, anything and everything green so don't spray it around plants you don't intend to get rid of.
Wait the time mentioned on the label before tackling the job of lawn replenishment. One thing about the use of RoundUP, use it according to directions and follow it exactly. Here, more is a waste just like using not enough.
Moisture: Lawns require moisture at the right time. Usually that is 1" per week, rain notwithstanding.
Fertilizer: Set up a fertilizing schedule with a high nitrogen type that will help grass grow and greens it up.
Read the directions on the bag.
To understand how much to spread, simply divide the first number (nitrogen) into 100--that is the amount per lb per 1000 square feet, to use. So if the bag contained 20/6/12....i.e. 20 percent nitrogen, 6 percent phosphurus, and 12 percent potash, then 20 into 100 = 5 lbs per 1000 square feet. If a bag contained 40 lbs, then it would cover 40 divided by 5 = 8000 square feet.
Your small lawn...200 sq ft could be properly fed with 1 lb of fertilizer. But read the directions on the bag.

If you live in a southern zone, then throw out the info on using a herbicide until you find out what it will do on your grass type and soil. Speak to your nurseryman before you do anything. Some types of southern grass cannot be given weed killers--it will kill the grass as well.

The first mistake you made though was in not watering--even what grass you did have needed moisture. The ground needed moisture. But as you describe, the ground needs to be tilled to improve drainage and its ability to grow grass

A soil test would not be a thing to avoid. There would be probably many soil labs that do soil tests but the one you can check on is at your local--the one nearest you--state funded university extension service. You can find their location in the phone directory under "STATE" government, list of colleges/universities....dept of agriculture...externsion services. Many large nurseries also do soil tests.
A soil test, if you do some research, can tell you what your ground needs in the way of chemical imbalance correcting. It can save you many dollars in experimenting with all what is usually given to a lawn to make it grow its best.
Even a 200 square foot lawn deserves being treated right.