Are those other plants (besides grass) really bothering you, or is it just that someone told you they're weeds and that you're supposed to be concerned? The profit-driven notion that a healthy lawn must contain only grass, or even more severely only one kind of grass, is not possible to achieve without chemicals or constant vigilance of patrolling and pulling out any other plants that sprout. Is that something to which you want to dedicate time every weekend? Will doing so improve your life in any way, or increase the enjoyment of the yard for pets, kids? IMO, the most healthy thing to do is mow whatever green things that are willing to grow there, without putting chemicals on the surface where you intend for your kids and pets to play. If something prickly sprouts, pull/dig it up. A mixture of plants will provide greenery for a longer period of time throughout the year.

Appreciate the response, thanks.

My goal is not to have that magazine ready picture perfect lawn. In the area that the weeds are the worst the yard quality, in the sense of bare spots and ability to hold up to the usage from the kids and pets. Grass at least seems to hold up better to a lively game of fill-in-the-blank.

They're far more invasive into our flower beds and garden due to prolific seeding.

Oh and I'm also allergic to dandelions, the whole plant, not just the pollen.

So between mud, more work in our other areas, and a itchy body I do believe they fit the bill for a weed in my book.

But I appreciate the tone, this isn't for vanity or keeping up with the Jones's.

I think you meant to say you don't appreciate the tone? I apologize most sincerely if what I said came across as chiding, I only meant to enlighten, theoretically. Doesn't help get rid of plants you don't want though, I realize that.

Many plants are called dandelion. If yours are Sonchus or Lactuca, they have latex sap, to which I'm also very allergic. I get a rash like poison ivy from latex sap, a more sympathetic ear can't be found. Is that the manner in which they bother you too? If so, after the cut ends dry up after mowing, the latex is not a factor, unless you like to roll around/roughhouse in the grass. That could cause ruptures in any leaves left, and the stem stumps, and would be understandably unacceptable, causing sap to get on skin.

If you have entire patches of them in what should be flower beds, you could smother them. Individuals aren't difficult to remove with a dandelion fork, but that is definitely hard on the back and knees. You might start with manually removing them from an area large enough to play? Keeping any other areas mowed so no new seed are dropped will prevent new plants from growing while you are trying to deal with certain areas, if interested in such an approach.

Bare spots in the sun aren't normal, generally. The first step would be determining why there are spots with nothing growing. When there is too much shade, grass can't grow well. Are your bare spots due to shade? If so, tossing more grass seed won't help fill in those areas.

There is a lawn forum, where you could get advice for creating conditions that favor grass but not taller plants like dandelions, and about non-chemical substances like corn gluten. PH level can also be a factor.

Whenever a pic (or several) is included in a discussion, the suggestions can be much more specific. I hope you find some solutions that are practical for you.

To get a healthy lawn that will resist what we call "weeds" you need to start with the soil. The University of Illinois Cooperative Extension Service does not do soil testing but they can supply you with a list of labs that do. What you need to know from that soil test is soil pH and the amount of major nutrients, Phosphorus, Potash, Calcium, and Magnesium, and the balance of those and what you need to do to correct any deficiencies. Then dig in with these simple soil tests to see what else might need to be done.
1) Soil test for organic matter. 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. For example, 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 drainsâ 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.

It will take time to get to that healthy soil and then grow a good thick lawn that will make it difficult for the "weeds" to grow, and this will not prevent all "weeds" from growing, but the thicker the grass is the fewer of those unwanted plants there will be.
Also mow the grass high, 3 maybe even 4 inches, because the more of the grass blade exposed to sunlight the more nutrients will manufactured to feed the roots which will then grow more grass.