To specialize in oncology or hematalogy/oncology you must first complete a general residency (ie - pediatrics or internal medicine) and then apply for a fellowship in onc or heme/onc. As far as I know, there is no such thing as just an oncology residency.
I am intending to go the route of pediatric heme/onc (thus 3 year peds residency + 3 year fellowship). The children's hospital that I worked at in Michigan as an undergrad actually hosted many DO residents, although it was an allopathic teaching hospital affiliated with Michigan State's College of Human Medicine (MD program). Infact, one of the attending physicians in the department of pediatric hematology/oncology is a DO.
I think the key really is to land a strong residency and do well in it. From there locate a good fellowship program and make yourself competitive for it. Most of the DO residents from the teaching hospital in MI landed competitive fellowships in a variety of fields.
Good luck.
j
