Glyohosate is the best all around plant killer and will not damage the soil (some people have planted seed within a day of use with no ill effects). Round-up is a common brand name.

Centipede is a FULL SUN type of grass. If you have pine straw and oak seedlings then you have trees in the area. Anywhere there is any shade the centipede will suffer badly.

I have oak trees in my front yard with St Augustine. I just mow over the oak seedlings.

Can you post a picture of your lawn to see what is driving you to rehab it? There might be some alternatives that become obvious to someone else here.

Sounds like you have shade issues to me. If so Centiweed is not going to work.

Thank you for the advice, St. Augustine is another one that I've been considering. The only reason that I was going to do Centipede is that I have some experience with it (although still a novice) and am a bit more comfortable with it. The trees that I do have aren't terribly shady, I know it may not be easy to tell in the picture, but this is what I am working with.

Sorry, I'm not sure why that picture came out sideways

Can't quite tell from the picture if the lighter trees are the source of your pine straw or not, but assuming they are since I can't see anything else in the pic that would be a source. That said, they not only affect your light, but your soil pH. Have you got a soil test done? I suspect there is going to be more to this than simply laying sod. Oh, and Oak saplings - agree with the rest, mow 'em!

Looks like you have heavy shade. That being the case Centipede has no chance. The most shade tolerant warm season grass is Saint Augustine, but from looking at the picture not even sure that is an option.

Between the fence and the trees I believe you have two choices: St Augustine or mulch. If you decide to mulch, dig out about 3 inches of sand and put it somewhere else. The fill with chipped/shredded stuff from a local tree trimmer. If they chip the tree branches at the site, that's what you want.

Otherwise get some flats of St Augustine. You do NOT want Floratam. Floratam is a great new variety of St Aug...for FULL SUN. In the shade it is a little weak. You might want to pull some of that sand out before you do the St Aug anyway. It looks like someone has brought in more sand over the years. For that small area I would use a push broom to move the sand off.

I don't see that a soil test is needed to tell you that you have sand. If you want to invigorate the soil/sand, make sure at least one of your annual fertilizer applications is organic. You can apply both organic and chemical fertilizer on the same day if you want to. There is no interaction or harm from that. The more organic apps you give it the faster the soil will become rehabilitated. The least expensive organic fertilizer now is alfalfa pellets (rabbit chow). Apply at 10 pounds per 1,000 square feet for your first app and then you can go with 20 pounds per 1,000 there after. If you apply 20 pounds the first time you may get a little bit of an odor due the the lack of soil microbe population needed to decompose the alfalfa.

And if you want to get back to your original question, you probably don't have to do anything to the soil in preparation for St Aug sod. Put the sod down and give it a few weeks to deplete the fertilizer that will have already been used on it at the farm. Or if you want to use alfalfa right away you can. Again, there is no interaction.

Here is St Aug care in a nutshell
Mow at the highest setting all the time. There is never any reason to mow at a lower setting (unless you get a dwarf variety).
Water deeply and infrequently. For that area I would use an oscillator sprinkler by the driveway set to water all the way over to the fence. Put some cat food or tuna cans out to see how long it takes to get a full can. That is your 1-inch time. Then start out watering that long. Watch the grass carefully to see when it wilts. When any part of it wilts, water immediately. This time of year it should go a full month without looking wilty. In the summer it might only go 7 days.
Fertilize (with organics in the south) on Washington's Birthday. Otherwise DO NOT wait until Memorial Day for chemical fertilizer. Most of the country can wait until Memorial Day but in the deep south it is too hot by then. Basically you should wait until after the spring flush of new growth is over and the grass starts to slow down. Then fertilize with either organics or chemicals.

Thank you all for the advice. dchall_san_antonio, That is some great information! I really appreciate it. I forgot to mention in the OP that I am planning on putting a little landscaping border around the trees and putting mulch in there.

Uh, that could be a problem for the trees. Tree bark should always be exposed to the air to absolutely ensure that it is not buried. The best way to do this is to expose the very tops of the roots. The roots don't have to be a trip hazard, but you should be able to see the part of the tree where the base widens (flares out) to form roots. Tree bark is subject to rot if it is buried under soil, sand, or mulch. Tree roots are not. If the tree rots around the bottom, it can kill the tree by girdling. The flow of nutrients is cut off at the base and the tree dies a slow death. So if you want to put 3 inches of mulch around the tree, dig down three inches below the root flare and back fill with the mulch.