Anemia for months. Iron meds not working. Should I get blood tests done?

Blood tests for iron. Yes, most definitely. It is time to see your physician for iron studies, hemoglobin electrophoresis, if indicated, B12 and folate (folic acid) levels, etc. You may need referral to a hematologist in the future if your anemia continues, depending on how anemic you are, cause, and symptoms.
Anemia. Anemia is diagnosed by a low hemogobin/hematocrit and other factors depending on the type of anemia. If you have iron (Fe) deficiency anemia, your Fe level is low and TIBC is elevated. Symptoms are paleness, fatigue, lightheadedness. You should get these studies repeated after taking Fe for at least 3 months.
YES. See a hematologist and get a CBC, CBC, reticulocyte count, serum iron, TIBC, B12 and folate (folic acid) levels for starters. I suggest oral iron and multivitamin supplementation until seen. Schedule an appointment this week.
Sure. Anemia can be Iron, Vitamin B12, Folate (folic acid) deficiency, Congenital manufacturing defects like, Thalssemia, Sperocytosis. It can be because of increased loss...bleeding, parasites in intestine. Lastly digestive problem, Liver, Spleen ,kidney, bonemarrow anomaly can cause. See PMD.
Definitely. Anemia that over a three-month time period of iron therapy does not respond should be investigated.