Use a wheelbarrow to transport soil, sand, or fill dirt around the yard. You can dump loads where needed and spread the material with a rake. For large areas, consider renting a small tractor with a front-end loader. Its scoop can move large volumes quickly.
A garden cart also works well for smaller jobs and is easier to maneuver than a full-sized wheelbarrow.
Renting equipment like a skid steer or small front-end loader allows you to efficiently move a high volume of material. These machines have buckets or scoops designed for materials like dirt. Just drive to where you need the material deposited and dump it out.
If you only need to move smaller amounts, a sturdy shovel can work. You may need to take more trips back and forth, but a shovel gets the job done without extra equipment.
For final shaping and spreading, a rake or hoe is handy. A landscape rake with a wide head can smooth and spread material efficiently. Consider renting a powered tiller or chipper/shredder if you need to break up compacted soil or blend in amendments like compost.