Growing lawns and gardens in clay soil can be frustrating because clay can form a barrier against roots, nutrients and water. Roots need to spread. Soil needs to drain. How to fertilize clay soils that can be nearly impenetrable is complicated. The short answer is that clay soils first have to be amended; a gardener adds material including fertilizer to the top layer of clay, loosening it to encourage plant growth and so that it can accept nutrients and water.
Test your soil to find out what kind of fertilizer you need to add. Different types of clay need different types of fertilizers and so do the plants you propose to grow. Most states have agricultural extension services that can test your soil or tell you how to get it tested.
Mix your amendment ingredients. There is no accepted formula for amendments to clay soil. Your mix should ideally be equal amounts of compost, pine bark, manure and gravel. All three organic materials are good, but if you only have one or two, use those. You should include gravel or shale to promote drainage. You can add non-clay, weed-free commercial topsoil.
Use your spade and shovel to break up the clay about 8 inches deep, about the length of a shovel blade.
Spread a 2-inch layer of your mix over the area. Break up clods of clay into ever smaller chunks. Work the amendment into the soil.
Spread a second 2-inch layer, adding the fertilizer or lime that your soil test said was necessary. Work this layer in the full 8 inches.
(Optional) As you amend your clay, it will become higher than the surrounding soil, helping to improve drainage. It will eventually settle some. To keep it higher, build a barrier around it.