looks like it already took over. Because it's a weedy grass in a lawn, most likely you'll need to get rid of everything and start over.

First contact your local office of the Clemson USDA Cooperative Extension Service and the can help identify that and can also help you get a good, reliable soil test which might do as much to control this as anything else. Many "weeds" grow in soils that are not good soils for turf grass and turf grass most often needs a good, healthy soil or it will not grow well. At the same time dig in with these simple soil tests, Structure. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. A good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top. 2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drains too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up. 3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart. 4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer you soil will smell. 5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy. to learn more about the soil you have and what yo need to do to make that soil into something that will grow a good, healthy turf. Here is a link that might be useful: Clemson CES

In your photo there are LOTS of little seedlings, which means there must have been lots of seeds in your lawn or in the compost you put down. You didn't mention the lawn having lots of weeds previously so I wonder about the compost. Where did it come from? Your options are limited while the new grass is just getting started since many broadleaf herbicides can damage the grass seedlings too. If you wait until the grass is established at least 8 weeks, then you may be able to use herbicides. Check the label of any product before using it on new lawns. I agree with kimmsr that correcting soil problems is the way to manage moss in a lawn. Moss is not a competitive plant, as you can see by the very limited root system it has. It grows in lawns where the grass is already doing poorly. This can be cause by lack of soil fertility (the most common cause in my experience), too much shade, heavy soils that drain poorly, and excess acidity (pH below about 5.2). Lack of organic matter isn't normally limiting for grass, but adding organic matter can improve drainage if that is the problem. Have you had a soil test done? You mention that the soil was determined to be acidic. What was the pH?

No plan to get them all. I try to keep the lawn low-maintenance and go on all natural. I may still spot treat the broad-leaf weeds, or just weed wack them. I still have to dig up the perennial grassy weeds, such as Johnson grass, etc....