I'm hardly an expert, having got our first house only about a year and a half ago, but I'll give this a shot. For reference, I'm up in Spring, so we should be in similar conditions. Your grass is most like St. Augustine, since that's what pretty much everyone here has. We've been trying an organic approach to taking care of our lawn. The backyard in particular was particularly weed-infested as the previous owners had a couple of large dogs back there that left big trails around the yard and there were a couple of overgrown mulberry trees we had removed that were dense enough to kill the grass underneath. Weeds filled these areas in quicker than the grass.
After doing our homework online, we've had very good results so far by mowing the lawn weekly with the mower on the highest setting and watering less. Letting the ground dry out a bit between waterings causes the weeds to die but the grass survives longer due to it's better developed roots (aided by mowing on the highest setting). I generally just wait until the grass starts to get a little dry feeling when walked on, and not springing back up quite as quickly, then water long and deep with the sprinkler. No weed-and-feed for us... At this point it's doing fine with no supplements other than the clippings we leave behind when mowing (we mulch, not bag). In the fall I'm planning on adding a thin layer of organic compost over the whole lawn to provide some additional nutrients and beneficial microorganisms and maybe renting a machine to aerate the lawn. Missed our short spring this year and didn't get it done then.
Our turf looks at least as good, if not better, than any of the other lawns in our neighborhood, and the weedy areas of the backyard do seem to have less weeds and more grass coming in now than before, so progress seems to be being made on that front too. We don't really have any problems with bugs in the lawn. I've noticed an increased lizard presence this year vs last that may or may not be related, and should help with bugs as well.
If you decide to go the chemical approach, be sure to read and follow the instructions on the bag as far as when/how often to apply and how much. Applying too many chemicals too often won't help your lawn, can actually hurt it, and results in more chemicals that can get into ground water, etc. Any weed killer / weed and feed products should have a list available of what kinds of plants they kill, so should say if they're effective against crabgrass.
Hope that helps...