Thanks guys! San Antonio actually was saved from most of the brunt of Harvey. It was supposed to hit us head on, but it veered East and we only got a few "tails" as they spiraled around. So just some steady rain.
Stuff that I bought:
I bought some Spectricide weed killer today that I plan on spraying out tomorrow. So hopefully that's a good start? I'm going to have to find a place here in SA to pick up the Celsius that you guys mentioned. If the Spectricide is a bad idea, let me know, I can take it back. I just figured I'd grab it in the mean time and ask here later tonight (I work nights and have downtime while here).
I also picked up a mower today as well - Toro Recycler 22 in. I kept going back and forth about that one and the Honda 21 at Home Depot. Hopefully it's a good buy. Seems kinda gimmicky with the fold up "SmartStow" function, but if it's a good mower, then I'll definitely use the fold up functionality in my garage (Which is full of stuff that I haven't unpacked).
Weed Killer:
http://www.homedepot.com/p/Spectracide-Weed-Stop-32-oz-Ready-to-Spray-Concentrate-for-Lawns-HG-95835-1/100352322
Mower:
http://www.homedepot.com/p/Toro-Recycler-22-in-Personal-Pace-Variable-Speed-Electric-Start-Gas-Self-Propelled-Mower-with-Briggs-Stratton-Engine-20334/100659631
To answer some questions:
I definitely want to help the St Augustine take over. I like the look of St Augustine better than the Bermuda, and my front lawn is actually pretty good with all StA. Is my only option to go out there and use grass killer to try and kill off the Bermuda? If so, I don't mind going that route, just would like consistent type of grass throughout.

Also I read or watched a youtube video or something, that recommended cutting the grass at the highest possible level to try and get the StA to take over the Bermuda... not sure how long that would really take though. :? I, like everyone, want instant gratification.

Front Yard:
The back corner that is in my original post is the SW corner I believe. There is some cover in that area from my neighbors trees too. I wasn't sure if it wasn't getting enough sunlight or if it's because of water/runoff or maybe something the previous owner did.
Thanks for the tip about the crepe myrtle trees. I actually have two, I'll have to take some better pictures. I going to start pruning some of the real low branches that are at the bottom of the tree. Wasn't sure about the top but I will have to do some trimming. The one in the picture below is actually pushing up against a fence and I think is what caused the fence to start leaning. See pic:

Also: Kinda curious what the other two trees are. I'll have to grab some pics of the leaves and try and find out.
As far as whether or not lawns go dormant? Not sure to be honest with you. I'd have to research it.
Pre-Emergent - Can you give me some examples of brands? Do I use this in conjunction with the Celsius?
Again, thank you. I know it's not a gianormous yard. I live right smack in the middle of Medical Center, San Antonio, and this is actually a pretty big lot here in this area and I want to make it mine.
Thanks guys.
