I wish it was mandatory that all posters had their location noted. You could live in desert or a rain forest. Makes a big difference. Since you mentioned St Augustine grass, I'll assume you're somewhere in the south or on the gulf coast.
St Augustine is a thick soft grass, so let's shoot for that. Bermuda can be thick and soft, too, so if you prefer that, I can give you a different plan. Besides, when you manage St Aug right, it will choke out all the other grasses and look great for you. Bermuda can still be weedy even when managed properly. The thin bladed grass with runners is likely bermuda.
If your landscape is level but uneven, I wouldn't worry about that yet. St Aug sod will add to the unevenness a little and correct it in some places. You can prep with compost and organic fert as was suggested by Sleeper. Or you can lay the sod now and apply the compost and fert on top. Be sure to roll the sod down with a sod roller no matter which way you decide to go. The roller ensures you have sod to soil contact necessary for roots to penetrate into the soil.
The rule for compost is to use 1 cubic yard per 1,000 square feet of area. Use a push broom to brush it off the grass blades. Even at the thin layer it can still smother the grass if left on the blades. If you sod with St Aug, you can use a sand/compost mix to fill the gaps between patches of sod.
The rule for organic fertilizer is to use 10-20 pounds per 1,000 square feet of area. Water it in to keep the birds from getting too much of it.
For long term maintenance, plan to water only once per week at the most. Water as much as you need to go allow the grass to make it until the next watering. If you make a mistake and the grass stresses before your week is up, water it to keep it alive but water deeper this time. Eventually you might have a grass with deep enough roots that it can actually go from rainstorm to rainstorm (yes, even in Texas) without watering. My neighbor is living proof of that.
For mowing, set your mower deck up to the highest setting for the tallest grass. This will give you the thickest, softest grass and it is the best for the St Aug. High mowing is the secret for weed free turf in St Augustine. You will still have to mow every week. There is never, ever, ever, never any need to lower your mower deck for St Aug. If you think there is, there isn't
If you decided you want bermuda, write back and I'll modify this last paragraph.