Pengh - welcome to the forum.
I tried accessing both LSU and Florida State Ag Centers but My computer is acting up and I was not able to. I wanted to find mowing height recommendations for St. A. In general using straight lawn fertilizer on a regular schedule (I think it is 3 times a year in our souther region) and proper mowing height and watering is the best way to have a lovely lawn. You do have a major problem with the weeds. I really don't like to use weed and feed but sometimes you have to. Look for a product designed for southern lawns and make sure St. A is listed on the label. Don't try to apply with one of those hand held spreaders. Invest in a good broadcast spreader. The bag of weed and feed will tell you what setting to use on the spreader. You will need to water it in. Again don't try to do that by hand. Set out a pulse sprinkler with a 360 degree spray pattern.
I have this argument with g every spring. He wants to weed and feed the entire yard. I don't want him using that in the back because we have a 150 year old southern live oak and my garden is in the back. The compromise is that he does his thing in the front and leaves the back alone.
that I can not access the info I want. The thing is that you keep the grass mown to a particular height never cutting more than 1/3 of the foliage. So during the warm rainy season you will need to mow more often - probably every 3 or 4 days. The grass will be healthier. The foliage will shade the soil and keep weed seeds from germinating and the weeds will not have a chance to go to seed. Fertilizing and good mowing practices will result in a lovely lawn - it just takes more time.
As soon as my compter stops being stupid I will post some links for you.