Best is to add organic matter, particularly compost. Be sure and add when your clay soil is dry, as working in wet clay is not only quite messy but will compact it even further. Peat moss is sometimes recommended as an addition, but this breaks down quickly in wet and clay soils, and can create an undesirable (for most plants) bog. Compost is good as it not only helps with the soil structure, but a compound (glomalin) the microorganisms (mychorrhizal fungi) in compost create binds the small clay particles together into aggregates with a waxy coating, thus creating more space between them for air and water to flow. As an aside, glomalin also benefits the soil, and earth in general, by storing carbon.
You can't really overdo the amount of compost added to clay soils. For a lawn and landscape that is a quarter acre, just increasing the soil organic matter from 2 to 3 percent would take 5000 pounds of an amendment! Don't get discouraged by this, as adding some is better than none. You can add organic matter over time, and you can deal with small areas or beds at a time. Adding 3 to 4 inches at minimum is recommended. If you have a local compost facility, check to see if you can have a bulk load delivered. This is cheaper overall and avoids having dozens of plastic bags to dispose of.
Best is to work organic matter deeply into the soil as roots will eventually end up there, and do so prior to planting. You can till as deep as a tiller will go. Or, dig sections of a bed at a time to a foot deep, work in compost, then replace the soil and do the next section. In future years, just work the surface, as organic matter you've worked in deeply will decompose quickly when brought up to the surface.