Later 20th Century records tend to be sealed (not always), but in some states adoption records prior to 1930 can be obtained from the court. It may be necessary to request the records in person. Prior to and into the early 20th century, adoptions were not necessarily formalized. In general, when adoptions are formalized, that happens in the appropriate court of jurisdiction for a given state -- the title of the court will vary. It may be called probate, Circuit, Surrogate, Chancery, Common Pleas or something else. Sometimes you can find reference to adoption in the names changes, which will be found in the local statutes prior to jurisdiction passing to the courts.
The adopted or foster child may or may not have assumed the surname of the family. Sometimes both names are used.. the birth name on more formal documents, the name of the family otherwise. In census records 1880 forward, the relationship often specifies adopted son or daughter. A marriage, death or other official record may reveal the name of the birth parents.
You may find children in orphanages, poorhouses or childrens' homes in census records, but the fact that a child is in an institution does not necessarily mean that both parents are dead.