So I want to transplant some trees in my yard from the woods that are going to be cut down?

If they're from the woods then they are probably huge, and if they are past a certain height and thickness, it's really not possible because of so many roots.

It is always best to choose a seedling tree to transplant. While a mature tree's root system is not always that deep, some young trees tend to put down tap roots. The top of a tree may be less than a foot tall, but the root may be twice as long. Check out the soil in these woods. You will have to deal with other tree roots while you dig down to free the young tree, but you may also have to deal with rocky soil. Smaller trees will adapt better and start active growth sooner than larger trees.

Furthermore, it's almost summer, and that is NOT the best time to move a tree. I recommend you put any seedling trees in pots, and set them in dappled shade (sunlight filters through leaves so there is a pattern of light on the ground) then into half day sun. Finding tall pots is a big problem. I found one place; see 3rd link. You can set them in the ground in the fall. If you live in the South, fall is THE best time to transplant most trees and shrubs. The roots keep growing through winter dormancy, so don't forget to water them if the weather turns dry.

This is no doubt a lot more bother than you were expecting, but there is a lot more to moving a tree than you knew. Saving native tree stock is commendable. Nurseries tend to offer limited species. Yes, it's good for the environment to plant anything.

It's a good idea, but if the tree trunks are more than a few inches in diameter, it will probably kill the trees anyway. If they are large trees, hire a machine to come dig up the entire root ball. This is expensive. If they are small trees, you can do it yourself, but you need to dig up the entire root ball. Dig a line around the tree below the edge of the branches (the roots extend into the soil at least as far as the branches). Then dig down at least 2 feet to get under the main roots.

Make sure you dig a hole large enough for the entire root ball. If any of it is perched above the existing soil after transplant, the tree will die.

Yes :) that's a great idea. I just don't know how you're gonna pull the tree up with the roots still intact.