With adoption, the genetic roots of an individual will be those of the biological parent. The genetics of the adopting family will play no part.
To work out the ancestry of an individual you can either follow the paper trail. This means that you will need to go to official records like birth, marriage, and death records, or censuses, to get information about each person and who their parents were. This may be difficult in cases of adoption and I am unfamiliar with the legislation in the US surrounding this.
The other alternative is DNA testing. When you do a DNA test, your DNA is compared against the DNA of the other people who have tested using the same company. The ethnicity estimates are then calculated based on how similar or different your DNA is to those other people. The actual ethnicity is then worked out from the self-reported ethnicity of the other individuals in the database. This works out all right statistically, but when looking at the results of an ethnicity test like the ones often sold by e.g. Ancestry or FTDNA then note that anything below about 20 or 15% could just be statistical noise (i.e. it may be random).
If a person is adopted and does a DNA test, and one or more members of their (unknown) biological family has also done a DNA test with the same company, then you may get something called a "match". The proximity of a match is measured in the amount of shared DNA between the two individuals. The more similar the DNA the two people share, the more closely related they are. Say, for instance, that my grandmother adopted my father and I did a DNA test and found that I had about 900 cM (centimorgans, a measure used in DNA testing) shared with a stranger who also had their DNA tested with the same company. I would then know that I'm pretty closely related to this other person, and I can contact them and hope that they respond and you can try to work out the relationship from there.
Although the above is predicated on finding matches in one company's database, many allow you to test with one company and then upload your data to other databases. This will increase your chances of finding matches that you're genetically related to.
In general, there's no mathematical formula to predict what the genetics will look like. It's not as simple as inheriting 50% each from your father and mother, and then 25% from each of your grandparents. You may inherit more from your father than your mother, and more from either one or the other of your grandparents than the other set. The genetic roots of any given individual is thus a mixture of all the DNA of that person's ancestors.
