I agree with everything animaux said, except possibly for bermuda
taking over in sunny areas, but this may be the case in Austin with
shallower soils. In northeast Tx., St. Augustine overuns everything
if watered enough. It requires more water than everything else, but
is more susceptible to diseases, which is one reason why it is better
to water deeper less often. Watering every day or every other day
causes a lot of disease problems.

If your yard has become dry enough for grass to die out, it is
probably compacted, so aerating is definately recommended. If it is
not a large area, or money is not a factor, this is a good time to
topdress with compost, but is not necessary.

Aerate, water deeply, fertilize, and mow at 3" and the st. Augustine
will spread out. How long? Sorry, depends on your conditions. You
can use plugs in bare areas to speed it up, but as animaux said, do
not patch sod, it will be uneven.

Good luck,
Bryan