Aerating clay soils is a short term solution to the problem. What that clay needs is organic matter, and once there is adequate amounts of organic matter in that clay there will be no need to aerate.
Moss is a first stage plant that Ma Nature grows where organic matter is insufficient to grow other plants in an effort to increase the amount of OM in the soil. I have had moss grown on my sand until I incorporated adequate amounts of OM into that sand.
Contact your local office of your Rutgers Cooperative Extension Service and inquire about a soil test so you will know something about the soils pH, nutrient levels, and percent of organic matter.

Here is a link that might be useful: Rutgers CES

If you had been leaving the grass clippings on the lawn for the past 3 to 4 years, you would have seen a lot more improvement. That is by far the easiest way to add organic matter to a lawn.

Start by mowing high and leaving the clippings. Don't till because that will just make a mess. Overseeding with white clover will help because it can do a better job of rooting in the clay.

The idea is to work smarter, not harder.

If you can find a source of granular humic/fulvic acid, get it and spread it on your clay soil. Heavily. Repeatedly. Your clay will turn into topsoil.

Thanks so much for the inputs, appreciate it! There is a huge area in my front and backyards that have no grass, just spots of weed and alot of thick moss. Putting down anything will need to be fully incorporated with the clay so i have no choice but to break it up right? Anyway i have started to turn the sections over by shovel today. Will start again tmrw then add the OM my soil needs. Hope to have a greener yard soon!

A source of humic and/or fulvic acid is organic matter, compost or other vegetative waste. Since what a good healthy soil might contain is 6 to 8 percent organic matter you should be able to calculate how much it would cost to apply some humic or fulvic acid to get to that point.
Mulch mowing, leaving the grass clippings on the soil, can add a small amount of organic matter to that soil but not, by themselves, enough to build up humus in the soil. Probably mulch mowing any tree leaves into the soil will not be enough either, it is not for my soil.

I get coffee grounds from Starbucks.
Last fall i was getting up 10-15lb + per day.
filled my compost bin very fast.

i added some gass clippings from my yard and the neighbors
+ a few food scraps here and there, and now ive got a LOT of compost.

you can even add a small amount of grounds straight to the soil, especially when you till it.

Some people dont like the idea of adding it fresh, because it will rob the soil of nitrogen to help break down the grounds,
but you dont need to add that much.

Fresh grounds attract worms, worms LOVE coffee grounds
left over coffee from the pot works OK too, but not nearly as good as grounds

worms will break up your soil and add nutrients as well

Humic and fulvic acids are the end products of the breakdown of organic matter. Their mild acidity chemically reacts with the alkalinity of clay and lowers the pH of the clay, making trace minerals available for the plant. Humic and fulvic acids also alter the structure of clay, breaking up the structure, allowing air and water to penetrate the soil more easily. Humic and fulvic acids also provide food for bacteria, bacteria that create the organic slime that condition the soil texture.

I used granular humic/fulvic acid on southern Idaho (Boise area) clay soil and it caused the soil to become the dark, crumbly, friable type of soil that grows healthy plants. The soil was deposits of clay from the Boise River. Try it before you criticize it. Just do a little experiment on some new garden soil.

Best regards, Paul

And to follow up on the question about incorporating material into clay soil; tilling is not required with granular humic/fulvic acid. Apply it to the surface and moisten it. Repeat.

Again, before a person criticizes this idea, I recommend that they try it. Be open to new ideas.

This is the first year of gardening this particular land. Of course it turned out to be CLAY. We left behind a great little garden that we ate from for 20 years. I am doing raised beds for the main vegetable garden but there are some existing spots I mulched and manured for things like the herb garden and the cool weather crops. I just started vermicomposting about a month ago so I am waiting for my first batch of castings. My cool crops are yellowing. I don't want to jump into the bag of fertilizer. I want to amend this soil. What can I do in the meantime? Amendments like mulch and manure take time. I don't want to lose these sprouts.

Has anyone heard or used a product from HumicGreen called Claymend? It supposedly breaks down the bond between clay much like what idaho_gardener (Paul) describes. If this thing works so well, what could be the drawbacks? Below is the link I found:

http://humicgreen.com/claymend/?gclid=CO6Vt6mZ170CFfNxOgodjksA-w

If anyone has had any experience with this product or something similar please share. The price per ft2 seems to be much cheaper than any organic or inorganic amendments.

Here is a link that might be useful: HumicGreen - ClayMend

Been a week since I turned 12'x60' section of my front lawn. Looks like I'm planting corn or something. Picked up EZ-Seed (or something like that) and laid it down. I also covered it with rolls of erosion mat but I fell short, need to get 2 more rolls tomorrow. Watered it and scheduled my sprinkler to water 4X/day for 20 min each starting at 7am. Is this enough, too much? I know I didn't add enough peat but I am hoping to get the grass going then put compost in the fall for the next 2-3 years or so to add more organic matter (pretty costly). Am I at least in the ball-park? Your inputs have truly given me hope that I can get my lawn a as green as a lawn should be! Thank you to all that have given their suggestions. Oh yeah, I didn't test the soil yet but will tomorrow. I figured I can always add fertilizer or lime or gypsum granules after sprouts come up. God I hope I didn't mess this up! :-)

There are no "magic elixirs" , other than organic matter, that will make clay soils workable. Get adequate amounts of organic matter in the soil and the humic and fulvic acids and the fungi that make those symbiotic relationships, the mycorrhiza, (myco = fungus and rizza = root zone) will develop without purchasing something that may not be what is expected.

Here is a link that might be useful: improving clay soils

I think you're doing the right stuff. Test the soil for needs, till in organic matter initially, and top dress with fine compost in subsequent years.

If compost seems expensive you might look for a bulk source where you can get a pickup load or have some delivered, rather than bags from the store. Some of that is not very good quality anyway. Does your city compost yard waste?

Thanks to everyone who commented on my clay prob. I knew this yard was crap, that is why I opted for raised beds and had a truck load of compost and soil hauled in. Trying to amend the existing plots that probably saw (at best) flowers and shrubs. I want everything in line with being sustainable. Robbed the worm bin early cause my cool weather crops are in existing bed and turning yellow. Added some blood meal too. I do compost outside. In fact I am hot composting a pile right now. Waiting for my 45 nitrogen. So between the outside compost and the worm composting I will get there. My husband keeps saying "But what about using Miracle Gro". Lord my day just keeps getting better.