Both of my heels feel bruised all the time. What could it be? Should I go see my podiatrist?

Have it evaluated. Heel pain is most often caused by plantar fasciitis, a condition that is sometimes also called heel spur syndrome when a spur is present. Heel pain may also be due to other causes, such as infection, tendonitis, arthritis, stress fracture, nerve irritation, retained foreign body, bone tumor, or a cyst. Have it evaluated by a professional and get the appropriate treatment.
Calcaneal bursitis . Calcaneal bursitis is a condition where a fluid filled bursa or sac presents on the inferior portion of your heel, separate from the insertion if the plantar fascia, see your local podiatrist for an x-ray and ultrasound for confirmation.
Yes. I don't know your diagnosis but by all means - a podiatric evaluation may provide the answer.
Yes. See a podiatrist as soon as possible. Possible causes of your pain can be bursitis, plantar fasciitis, faulty foot mechanics among others. A podiatrist can thoroughly exam you, take necessary x-rays and get you started on your heel pain relief.