Easy - seriously. Your lawn is in rather good shape compared to mine. I moved to a weed patch with about 10 square feet of real grass in 2014. Now there's upwards of 1,000 square feet of real grass. I'm going to give you the plan to convert this to 100% St Augustine over time. If you want something in a hurry, let us know and we can get you on a different plan. If you want an immediate lawn for a January wedding or something, the best grass is grown on grass farms in Wharton - just a hop, skip, and a jump from you, so it will be as fresh as can be if you want to go that way.

Water it deeply about once a month starting now. Put out some cat food or tuna cans and time how long it takes your sprinklers to fill the cans. That's how long you will be watering from now on, so write down that time. My time is 8 hours, but I use extremely slow sprinklers (turbo oscillator). Yours might be as little as 20 minutes, so keep an eye on the cans. I'll put more about watering at the bottom of this reply. Proper watering is the most important part of having a good looking turf, so pay attention to this.

If you want to have a St Augustine lawn, then raise your mower all the way to the highest setting. You have it on the lowest now. You can mow the weeds off in March and the St Augustine will catch up in April. For next spring I would not spray any weed control. Just having something green is much better than bare spots where the herbicide killed off the weeds. Some time about late March the grass will awaken. Once that happens it will leap out of the ground like it's on rocket fuel. Let that happen. Keep after the mowing on a weekly basis. After a few weeks it will slow down to normal growth. At that point, I usually back off to mowing St Augustine every 2 weeks. That allows it to get taller than the surrounding grasses. The tall coarse bladed St Aug will provide too much shade for the grassy weeds like bermuda to take hold. By July the parts where the St Aug is doing will will become very dense. St Augustine will spread about 15 feet per year in all directions, and uou have enough now for it to take over in one season.

At this point I would use an organic fertilizer. I'm assuming you have not been on an organic program. Call your local feed stores to get their prices on corn meal, corn gluten meal, and alfalfa pellets. If they have corn meal for less than $10, use that. If they have corn gluten meal for less than $20, use that. If they have alfalfa pellets for less than $12, use that. Those come in 50-pound bags. The application rate is 15 to 20 pounds per 1,000 square feet. I would apply by hand rather than a spreader. If you miss some spots you can always add more with organics. You likely won't see anything happen this year no matter what kind of fertilizer you use. This application is for next spring, but do it now. In the spring don't fertilize until Memorial Day. Trust me on this. It will cause more issues if you fertilize early. Really after this fertilizer drop now you don't have to do anything but water and mow in the spring until Memorial Day. On Memorial Day you can use just about any fertilizer you want. I like organic, but it's up to you.

Watering: Deep and infrequent is the mantra for watering. This is for all turf grass all over the place. Deep means 1 inch all at one time. Put some cat food or tuna cans around the yard, and time how long it takes your sprinkler(s) to fill all the cans. Memorize that time. That will be the time you water from now on. My hose, sprinkler and water pressure takes 8 full hours to fill the cans. Your time will likely be less. I like gentle watering. As for watering frequency, that depends on the daytime air temperature. With temps in the 90s, deep water once per week. With temps in the 80s, deep water once every 2 weeks. With temps in the 70s, deep water once every 3 weeks. With temps below 70, deep water once a month. Note that you have to keep up with quickly changing temps in the spring and fall. This deep and infrequent schedule works in Phoenix and in Vermont, so it should work for you. The reason for deep and infrequent is to grow deeper, more drought resistant roots and to allow the soil to dry completely at the surface for several days before watering again. If it rains, reset your calendar to account for the rainfall.