What causes foot pain as I age, especially when I get up from sitting or out of bed?

Multiple reasons. A common complaint of patients with plantar fasciitis is morning pain when first stepping out of the bed. Arthritis pain can also be worse before getting "stretched" out in the morning. A long list of other things cause foot pain.
Contractures. Oftentimes people get pain to the bottom of the foot or to the achilles tendon area. This is a result of the plantar fascial ligament on the bottom of the foot or achilles tendon in the back contracting and then when you go to step and use them they are stretching. Often times, a splint is used to keep the ligament or tendon stretched so they don't contract in the first place.
Fasciitis, arthritis. Sometimes pain in the am is plantar fasciitis. It can also be systemic arthritis like rheumatoid arthritis.
Fluid dynamics. When you get out of bed or get up from sitting, similar things are happening in your feet. Areas in your feet that are already inflamed now begin to receive more blood your heart due to your standing as well as veins not clearing blood out fast enough. With your first steps, the added pressure cause pain. Further steps causes muscle contraction and deeper breathing reducing pressure and less pain.