Hard to have a nice yard, when neighbors have a Dandelion Nursery!

pishta said

Imagine a sea spider but it's a weed. It grows about 1/2 inch up then makes a 90 and then grows out in every direction. It'll get about 36" in diameter and you just get a shovel under it and find its center, then you pop it up ( maybe 6" single root) and throw it away. The other spreading plant like this one has a red rubbery flower like a succulent and is the same growth pattern, find the center and pop it out. Kinda like a red/purple rhubarb branch.

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Sounds like purslane. Hard to eradicate, but relatively easy to control. It likes poor soils, fix the soil and other stuff will crowd it out.
Don't let it get big or get to the flowering stage, its a prolific seed producer. Goes great in salads though, and makes a great compost additive (until it flowers). We find it's a good mulch in the garden and just let it spread. It's resilient to being walked on and keeps the soil moist beneath. It's not a heavy feeder, and helps to break up hard soils.
We've noticed that tends to thin out or disappear once the soil improves and other plants or mosses take hold. Purslane doesn't like a lot of competition, and is easily shaded out by taller plants.
Amend with compost and organic matter, and do a soil test to help figure out what your ground lacks. Compost and mulch will go a long way to helping grass seed take hold.