Additional tests include: hemoglobin electrophoresis (for beta thalassemias) and direct antiglobulin test (DAT or Coombs' test) to distinguish a hereditary spherocytosis from an immune mediated spherocytosis.
As Dr Blanchard suggested, for thalassemias : first you have to do a full blood count, look at the blood film, then proceed from there to do Reticulocyte count and Hb H preparation, HPLC for Hb A2 and F and you may need Iron studies. For spherocytic hemolytic anemia you need to check the patient history and family history, FBC, blood film, Reticulocyte counts, Direct Coomb's test, cryohemolysis test maybe a useful test if Hereditary spherocytosis is suspected.