Many "weeds" grow only in soils that are not good soils for turf grasses, although many others have adapted to a wide range of soil types. Given that then the first thing you should do is contact your counties Cornell Cooperative Extension Service office about having a good, reliable soil test done to see what your soils pH is and what the levels of P, K, Ca, and Mg, are. Then maybe dig in that soil using these simple soil tests,
1) 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 your 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 see what else needs be done to make that soil into a good, healthy soil that will grow a good turf.

@Kathy: Sadly, I have no suggestions, just commiseration. I'm in about the same boat here, and also feeling overwhelmed. I feel like I could be at it until the cows come home if I try to dig all those weeds by hand, and the dang things just keep coming back!