I want to Transplant some young cottonwoods that are growing in our roadside ditch to our yard.what is the best way to do that

Are you sure you want to do that? They can be really messy when they are blowing their cotton around. We have them all around our yard and it looks like it's snowed in the middle of summer sometimes. Many people have allergies - I can't mow the lawn when it's white without sneezing my brains out. They also require lots of water, so you should make sure your water table is pretty high where you want to transplant them to. They also send up shoots out of the lawn from their roots when they get bigger.

I have found them easy to transplant. I waited until they were going dormant, loosened the soil around them with a fork and shovel, and sort of pulled them up by the roots. They all lived, grew quickly, and became a good erosion control for a ravine on our property.

@Clint . Male cottonwoods do not produce the cotton seeds.
Which is good news. The bad news is you can not tell the sex of the tree until it is older and large. The wind blows the female tree seeds far and seeds will clog air conditioners. The yard will grow lots of little trees. Cottonwoods are water lovers. They will put out roots in access of 40 feet looking for water, be careful of foundations, water, sewer or septic lines. They are excellent trees to plant in areas that accumulate standing water. Not good for hot and dry areas.
May you be blessed with a wonderful week!

with all that's been said so far. Most important, if you want them to survive transplanting, you do not want to do it at the height of summer.

Wait until they are going dormant, as porkpal said - that is when the leaves are turning color and starting to fall.

Above all don't plant them within 60ft. of your house. There are much better landscape trees. In the fall, your local nursery will probably have some beautiful trees marked down to move out and fall is a good time to plant most trees.

Weed tree here in the Pacific NW, but to each their own. Good for eagles to perch on, taller than anything else. They do like wet feet - all of mine (none planted) are close to running water. Good luck.