Answer: To transplant a small tree, dig it with as large a rootball as possible and replant immediately at the same depth that it grew naturally. The soil may all fall off in which case just do the best you can. Water it in well. Mulch with several inches of organic mulch. Try to transplant them to a spot with similar growing conditions as where you find them, and make sure the soil stays moist until winter. To be honest the smallest trees and those without tap roots will move the best and evergreens are normally transplanted in the spring because it is less risky, so you may do a lot of work for nothing.
You might also want to consult with a professional arborist and/or a professional landscape architect about protecting the larger trees on the propertyboth during construction and afterwards. Tree roots cover a larger area than their branch spread and are very sensitive to changes in grade, changes in surface drainage patterns, and soil compaction. In addition, you will need to do what you can to make sure that the existing trees are protected from inadvertent construction damage.
Good luck with your new house!
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