The quickest way to a nice green lawn is sod. However, in my first house, the builder planted grass seed and with care we had the greenest lawn on the street!

First you need to define what your expectations of your grass.
Golf-lawn - expect to pay alot every year for various chemicals, the time to apply them throughout the year, and water weekly. Of course the first major drought will kill the grass and you will need to start all over.
Weed-lawn - no time or expense, and they thrive in a drought.
Nice green organic lawn - The organic forum should have a nice faq about this, but in general you only have to mow to 3-4 inches (instead of the 1-2 inches in the golf-lawn). The look is a bit like a shag rug, but it is very green, very thick, and does not take much care or expense.

True story - our neighbor had a beautiful shaggy grass lawn. They sold, and the new owners mowed it 1-2 inches. The lawn just about died. It still looks terrible, even though the new owners try all sorts of expensive treatments.

Me personally, I don't have the time or the money, so we pretty much have a weed lawn, and try to keep it at 3". The lawn is not a priority for me. But you need to decide for yourselves what you want.

Zoysia grass is an option, if you don't mind the brown in the winter.