I am on my second home where the entire property is composed of clay fill. This time it is about 4 acres. After trying everything out there, I have come to the conclusion the only reasonable option is to plant the grass in the clay, you may not have quite as nice a lawn but, it is possible to grow very nice grass in clay. You just have to feed it more.
For the garden, the least complicated way I have found is to add commercial compost. In this area, this tends to be made from decomposed mulch and mulch byproducts. It is effective, and contains no weeds, (I consider this a big bonus). It does however, have a strong odor, much like you get with some mulches. This can last a few weeks.
If you have a slope, tilling in anything can be a problem. Particularly with erosion. I would consider a slit seeder and sow in the soil as it is, if this is a concern.