Definitely start with a soil test. You mention that the soil is black, rich, but artificial fertilizers and many pesticides can lead to biologically poor soil. Without good soil biology, much of the nutrients you dump on your lawn remains unavailable to grass. If you can afford it, get a bioassay done (like that from http://soilfoodwebnewyork.com/ but from a local lab).
You also mention that your soil is moist. This in itself can be the problem. The number 1 cause of lawn problems is improper watering. Lawns should be watered deeply and infrequently, and you should let them dry out between watering. This encourages deep turf root growth and helps it compete with weeds. You may also have drainage issues - which a soil test may be able to detect. Continually wet soils lead to disease and prevent turf from competing with weeds effectively.
Finally, I can't tell from the photo exactly, but it looks like your grass is cut really short. You should not cut your lawn below 3" during the growing season, and never cut off more than 1/3 of the height of the grass. Taller grass has much more leaf area, shades out weeds and their seeds, cools the ground below it to reduce evapo-transpiration and heat stress, and leads to much healthier lawns.
Finally, leave grass clippings in place if at all possible. These clippings can be an important input of both carbon (food for the soil biology) and nitrogen (food for the turf).