There is a short window on the Gulf Coast, roughly mid-February to mid-March, where a yard is dormant enough to work on and close enough to waking up that the work pays off immediately. Do the reset then and the rest of the year is maintenance. Miss it and you spend until June catching up.
Start with the beds, not the lawn. Cut back the perennials and ornamental grasses that died down over winter, pull the weeds that got established while nobody was looking, and clear out the leaf litter that has been sitting since November. Doing this before the new growth starts means you are cutting into bare stems instead of trying to pick dead material out of fresh shoots.
Then re-cut every bed edge. This is the step people skip, and it is the one that does the most visible work. A crisp, deep edge between turf and bed makes an ordinary yard look maintained even before anything is planted, and it keeps the St. Augustine from creeping into the beds for months.
Now mulch, at three inches, pulled back off the trunks. Then move to the lawn: clear the debris, and take the first cut slightly low to remove the winter tips — this is the one time of year a lower cut helps. After that, raise the mower and leave it up for the summer.
What to skip: do not fertilize yet. The grass has to be actively growing and fully green before feeding does anything useful, which in most Houston yards means early April at the earliest. Feeding a dormant lawn in February is a well-timed gift to the weeds and nothing else.