For most cases, don't try to remediate the soil during the winter if your soil freezes. Products applied to frozen soils might not travel deeper into the soil and get washed away. Try to focus in the months the grass is actively growing. Also avoid applications in the peak of summer under drought conditions since it might lead to more plant stress. This is a marathon and unless you are deficient (rare) most of the amendment we do is to get the soil to an ideal condition.
If pH is less than 6.0, then it will need lime. Your soil test should tell you how much lime to apply per ksqft. The guideline is no more than 50lb/ksqft every 6 months, unless you are using a fast acting lime. For a fast acting lime, follow the bag recommendations. There are 2 types of lime, dolomitic and calcitic. Dolomitic limestone is a mix of calcium and magnesium; while calcitic is mostly calcium. Most soils do not need magnesium, too much can cause the soil to become too hard and it cannot be removed from the soil. Unless the recommendation was to also increase the magnesium in your soil, use calcitic to increase your pH. For nitrogen, a low pH soil will benefit from using ammonium nitrate (34-0-0) at no more than 3lb/ksqft/month.
If soil pH is above 7.0, then the best way to reduce it is with elemental sulfur at 5lb/ksqft per hot weather season (once a year in northern states vs twice in Houston, TX). Why? Elemental sulfur needs a microbial breakdown to convert it to sulfuric acid and release the hydrogen which is what reduces the soil pH (more info in this Ohio State article). Too much elemental sulfur without the microbial activity can lead to accumulation and too much being released at once (hurt your lawn). There are other ways to reduce soil pH and this Purdue article describes it and this Iowa State article.
But (there is always a but), lowering pH in the soil might not be possible/cost effective. If the soil has enough calcium carbonate, the quantities needed of elemental sulfur are too much. This article describes a study that tilted 240lb/ksqft of elemental sulfur. It dropped the pH by 0.5, but it increased over the years. Continuous application might keep it there (but at 5-10lb/year?) and using other techniques (citric acid) will help.
Triple superphosphate (TSP) 0-46-0 - Apply at 2lb/ksqft every month the soil is growing
For low pH soils you can use:
Diammonium phosphate (DAP) 18-46-0 Apply at 2lb/kqft but account for the nitrogen being applied 0.36lb N/ksqft. It will slightly increase the pH.
For high pH soils you can use:
Monoammonium phosphate (MAP) 11-52-0 - Apply at 2lb/ksqft, but account for the nitrogen being applied at 0.22lb/ksqft. It will slightly decrease the pH.
Sulfate of Potash (SOP) 0-0-50. Apply at 2lb/kqft. This will also provide some sulfate sulfur the plant needs too. It can be hard to find. Go to the hometown folders in the forum to ask someone local to your state for sources.
Muriate of Potash (MOP) 0-0-60. Apply at 1.6lb/ksqft. MOP can be harsh into the soil(chloride), but it is widely used by most fertilizer blends (eg. Scotts). Try to find the SOP since it is better, but don't sweat it.
Potassium Nitrate 13-0-44. Apply at 2.2lb/ksqft, but account for the nitrogen content of 0.28lb/ksqft.
For soils below 7.0, you can also use FAS. You can also use any other source of iron per the bag rate. Avoid applications with moist/wet lawn, since it can be absorbed via the leaves and cause a black lawn. After sweeping the concrete, run your irrigation after application to clean the leaves if the lawn was moist.
This article from Harrells is a great read on soil iron in general. The irony in iron
Iron stains. It will stain your concrete driveway, patio, etc. Be careful how you apply it and where you mix it. Chelated sources of iron stain less or not at all.
Applications rates (maximum per month)
10-10-10 - 10lb/ksqft
12-12-12 - 8.3lb/ksqft
13-13-13 - 7.7lb/ksqft
16-16-16 - 6.25lb/ksqft
20-20-20 - 5lb/ksqft