here's that weed. Its the littler green in the picture of my lawn

Plan (but first the rules)

1. Use fertilizer to fertilize.
2. Use herbicide to kill weeds (or pull them)
3. Don't use weed n feed for anything
4. Don't use any of the above in the summer heat

The brown grass might be dormant Kentucky bluegrass. Let's hope it comes back as soon as you get regular water on it and the temps cool down.

The viney weed looks at first like spurge, but I suspect it is something worse. I have not seen it in San Antonio, so maybe it is a northern weed?? Hopefully bpgreen or tiemco will come by to comment. My first attack would be with Weed b Gone Chickweed, Clover, and Oxalis. Once the daytime temps are below 85 you can use that. Spot spray only the weed and do not drench the soil. Most herbicides are a contact type chemicals.

How are you watering? Frequency and duration?
In a month you can fertilize again.

Are you concerned about the yellow looking grass?