The short answer is to add more organic material, particularly compost but there is more to the story. Soil particles come in different sizes, which contribute to soil texture. Sand particles are the largest, clay particles the smallest.  Being so small, clay particles pack together not allowing the necessary spaces between them for air and water to flow, especially air that most roots need to function. Contributing to this is the fact that clay particles are plate-like, stacked like a deck of playing cards, with little space between them. This traps water for long periods. One positive side to clay is that its particles hold onto nutrients, making them more fertile.

So how do you increase the space between these particles? Some recommend adding sand, but unless it is coarse sand, and you add about 3 parts to each part clay soil, the soil structure will likely just get worse. When planting, some recommend putting gravel
in the bottom of the planting hole. This is bad as it just raises the water level, creating what is termed a"perched water table". Another recommendation is to add gypsum, similar to lime, but this may increase calcium and pH levels too high (clay soils tend to be alkaline).

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.

You can't really overdo the amount of compost added to clay soils. 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.

Another method to increase organic matter in clay soils is with cover crops.  These are crops planted as you prepare beds, for a season or year prior, or in fallow periods between annual crops such as flowers and vegetables. They consist of small grains and grasses like buckwheat, ryegrass, and oats. Legumes, such as clover, also benefit by "fixing" nitrogen from the air for use in the soil. Cover crops have additional benefits such as suppression of many weeds.

If you are planting vegetables, till or break up with a spading fork the area you'll plant. Then build a berm, or raised bed with sides 6 to 12 inches high or more, over it and fill with a good loam topsoil.

If you're planting an individual tree or shrub, it is especially important to chose ones that will tolerate clay soils. Their root systems are so extensive that you can't amend the soil well enough over a large enough area, especially for trees. Dig the hole only as deep as the pot or rootball. Otherwise it will sink over time, causing the plants to end up too low. This is a major cause of woody plant demise after a few years.

Don't make flat sides to the planting hole that wont let water drain. You'll only be creating a bathtub for the roots, without a drain. And only amend the backfill soil with no more than one third of a better soil as the plants were growing in.

If planting large plants, break up an area around where you'll plant (out to as wide as the plants will eventually grow) with a long spade or fork. Add organic matter on top of this area which, over time, will work into these cracks you made.

For low, groundcover perennials that tolerate heavy clay soils, consider the carpet bugle or dead-nettle (Lamium), both of which can spread vigorously. Low to medium height perennials, mainly attractive for their foliage and good for massing, include the lady's mantle, sea thrift, pigsqueak, daylily, Japanese iris, Japanese painted fern, Ostrich and Cinnamon ferns, and the
late-spring flowering globeflower. Most spring-flowering bulbs need well-drainedsoil, however, checquered lily (Fritillaria meleagris) will tolerate clay and periodic wet soils.

Taller perennials for clay soils include aster, Helen's flower (Helenium), goldenrod, goatsbeard, foxglove, coneflower (Echinacea), false sunflower (Heliopsis), blazing star, black-eyed daisy (Rudbeckia), meadow rue, Joe-pye, and the compass plant (Silphium).

There are actually many shrubs that will tolerate clay soils, some better suited to them. These include the chokeberry, Siberian
peashrub, shrub dogwoods, forsythia, common ninebark, potentilla, currants, shrub willows, and viburnums.

Best choices for smaller trees include European alder, river birch, hawthorns, crabapples, and ornamental pear. For large trees consider hickories, hackberries (a native plant and vase-shaped replacement for elms), black and green ash, common honeylocust, Kentucky coffeetree, butternut, larch,
amur corktree, cottonwood or aspen, oaks, willow, linden, and elm.

Following are links to university websites with more info:https://forestry.usu.edu/news/utah-forest-facts/gardening-in-clay-soils

https://hgic.clemson.edu/factsheet/soil-conditioning-establishing-a-successful-gardening-foundation/