What possible reasons would my second toe on my right foot be numb, along with the side of my great toe? It has been this way for at least a week.

Varies. Sounds like an interdigital nerve could be inflammed. This can be due to a number of reasons (trauma, tight shoes), and will likely resolve without treatment. If it persists or gets worse then a podiatrist may be helpful.
Diabetic new shoes n. Are you diabetic? Are you wearing new shoes? Any injury? Sitting in a new chair? You can ask you pcp to refer you to a neurologist.
Nerve irritation. The key is finding out which nerve may be irritated. You can have a peripheral neuropathy or even an irritation from a nerve in your back (radiculopathy) even though you may not have pain in your back. The toes you describe can be associated with the L5 nerve in your back. See a board certified pain specialist for help.