Here is one way to plant potatoes. It assumes that the ground is clear of weeds and reasonably free of stones, and that all the usual preparation such as spreading and incorporating compost and chitting of tubers is done and sorted. I focus here on getting the tubers into the ground. If all the muck spreading was completed in the fall, by spring time the ground will be somewhat compacted.
I use a garden line to ensure a straight row, plus a pointed shovel. The garden line suffers quite a bit of abrasion from metal tools of various kinds, so it is made of something stretchy and durable such as braided nylon. The shovel is a short D-handle type. A garden rake for smoothing off the rows, a bucket or other container with the seed potato pieces, and a second bucket to store a small quantity of soil.
My own preference is to get the tubers into the ground as soon as it can be cleanly worked. The frost has to be completely out of the ground. If the soil is too wet it will stick to your tools and your boots and it will be a most unpleasant experience, so I wait until the soil is sort of dry on top. Planting early uses time when little else can be done. Delaying puts potato planting into a timeframe when a gardener is busy with other things and can ill afford the time. Inevitably there will be some damage from late frosts which nip emerging shoots and turn the green shoots black. The tubers recover very quickly; the damage can be somewhat mitigated by early hilling of the soil over the emerging shoots.
Decide where the first row has to be. Stretch the line the length of the proposed row. Allow for the fact that as digging progresses the tuber will be in the soil a few inches to the side of the line. Starting at one end of the row, stand on one side of the line, push the shovel into the ground on the opposite side and remove sufficient soil to allow the seed potato to sit comfortably at the required depth. Set the soil removed to one side, say in a wheelbarrow or bucket. This soil will not be required again until the last hole is dug.
With the first tuber in the ground and still uncovered, move to the next position along the row and remove the soil, using this soil to cover the first tuber. Continue along the row using the soil from the current position to fill in the hole left at the previous position. When the last tuber has been planted, take the soil saved from the first hole stored in the bucket or wheelbarrow and use it to fill in the last hole. Finally smooth off the soil left rough by the planting, making sure to leave the rows plainly visible.
The same technique can be used to plant other types of tubers or corms, such as gladiolus.