You can improve any soil -- clay, sand, and everything in between -- by adding organic matter such as shredded leaves and compost. Don't bury it too deep -- six inches is fine. Or apply it to the surface and let eartworms dig it in for you.
You can also improve some clay soils by applying gypsum. A faster solution is to garden in raised beds. The beds need be only about 8 inches high for annuals, 12 inches high for perennials and 18 inches for shrubs and small trees. Fill with good quality topsoil purchased in bulk from a landscaping company. Inspect it carefully before you buy it, because there are many poor-quality soils sold as topsoil.To find well-adapted plants, search Google for "trees shrubs clay" and you'll find a number of lists. Choose the one that best fits your region.