welcome to the wonderful world i share with you! This was my 1st attempt @ a veg. garden and it actually didn't go to bad considering I read up all winter long in preparation for the growing season.
I tilled lots of store bought compost from lowe's into the rows, I feel this helps greatly cuz my soil wasn't even growing weeds!!!then i piled on dead leaves from last fall, lotsa straw, coffee grounds, etc.

I been reading into cover crops, and forage radish...i think it was......was supposed to be great for breaking up heavly clay and it's subsoils....dunno if i'll try it or not, might just mix some winter rye with a cover crop mix.....

also in the mean time, you could be working on a compost pile, seeing how you got access to hay and manure, this will also greatly help the soil whether you till it in or not. plus adds nutrients, etc. good stuff.

What your soil needs is lots of organic matter. Manures that have not been composted, to help eliminate the disease pathogens that may well be present, should not be put into a garden soil that will be used to grow food, but compost is on good source of organic matter.
What organic matter is fairly readily available at very low cost in your area? If there are lots of deciduous trees then shortly there will be lots of leaves readily available since many people, still, think of these as a waste product to be gotten rid of as quickly as possible and they will do your soil a lot of good.

I would suggest starting a compost pile right in the middle of your garden plot. Use the manure, fall leaves, kitchen scraps, sawdust, basically anything that once grew. Make the pile as big as you think you need to cover the plot with a couple of inches of compost. Then make the pile twice that size. Mix every thing together pretty well while moistening it. By spring, you'll have some great stuff to fork/shallow till/surface spread on your plot.

One of the regular posters here has a list of 200 or so items that can be composted.

You may have to be patient, the soil may be very hungry and the compost may dissapear quickly. That happens to me with every new gardening bed I put in.

You may also wish to do a search on lasagne gardening.

also, another way to add organic matter, mulch with straw.
if you are seeding a lawn you'll probably mulch with straw anyway, you can put a layer of straw down around a plant after you plant something and maybe even then use wood mulch for an even longer lasting source of organic matter for your plants

and if you plant anything, like kimmsr said, find organic matter readily available like bagged compost from garden store, etc. mix this into the soil of your planting hole: good for soil tilth, plant health, micro organism growth.

compost is revered to be one of the best tools for gardening and i can tell you from this being my 1st year of successful and major gardening, that it does help.

my personal goal is to first get my soil where worms will actually live in it, but i'm not going to stop there.

When I first moved here, I hated the clay. I had come from a delta area of loamy soil. I hurt my arms and shoulders digging with a pick axe. I learned over time how to work with the clay. First, only work it after it has rained and you've waited a couple days. It will be like moist chocolate cake. Don't work it when it's real wet, it will clump and be like pottery clay. Dig your planting holes wide. As deep as the plant needs to go, but twice as wide as the rootball. Add organic matter like compost, and mulch mulch mulch. I agree you can start a compost pile, and/or use the hay to mulch around whatever you plant. I generally do the cardboard/newspaper lasagna gardening or sheet composting method. Clay does have alot of good qualiteis like nutrients and water retention capabilities.
Laurie