Pair iron rich foods with foods high in vitamin C. This will help increase absorption.
Avoid pairing iron rich foods with foods high in phytates and tannins (iron absorption inhibitors). These are some examples:
Red wine, coffee, tea
Spinach, chard, beet greens, rhubard, and sweet potato
Whole grains and bran
Foods that are iron absorption enhancers:
Oranges, cantaloupe (musk melon), strawberries, grapefruit
Broccoli, brussels sprouts, tomato, potato, green & red bell peppers
Veg*n foods that are high in iron:
Enriched breakfast cereals
Cooked beans and lentils
Baked potato with skin
That said, your hemoglobin is low, so there is nothing wrong with taking a supplement. From my understanding, when you're anemic, there is less blood to circulate through the body, which makes your heart work harder to get oxygen to all of your body's cells.