Instead of seeing clay soil as a negative, it is not very difficult to change the soil slightly and utilize its beneficial properties to enhance your garden. Clay soil retains many minerals necessary for healthy plant growth, and it retains water quite nicely. The main issues you will need to address are its texture and possibly its pH level. Once you address those, the more you garden the healthier your garden will become--naturally.
Test the pH level of your clay soil. Follow the directions included with your soil pH test kit, as they differ from manufacturer to manufacturer. This will tell you whether or not you need to add some gypsum or lime (which aid with calcium-deficient soil), bone meal (which adds phosphorous), or sulphur (which corrects excessively alkaline soil).
Dig up your garden full of clay soil with a shovel, down to a depth of around 6 inches. Break the soil up as much as you can using the shovel so it crumbles between your fingers if possible. If it is too wet to break up easily, wait until it has dried out a little more before continuing.
Work compost and any necessary soil amendments (as determined by your previous soil pH test) into your soil using your mattock, hoe, spade or other digging implement of choice. If you are using commercial compost, it will tell you how much you should add. If you are using homemade compost, the recipe you use should tell you how much to add. Any amendments you need to use will similarly tell you what and how to apply them.
Let your garden rest for at least a week before planting. This will allow the pH levels to balance out and will allow the soil's natural ecosystem to adjust to the changes you have made. If you plant sooner, you run the risk of endangering the root systems of your growing plants.
Repeat this process every growing season. You should notice a marked improvement every time.