Learn German: Der Fuß tut weh.

The infinitive verb is wehtun, which means to hurt. Because it is a separable verb, the core verb tun takes the second position in a normal statement, and the prefix weh gets sent to the very end of the sentence.

Is it more natural to say Der Fuß (the foot) instead of Mein Fuß (my foot)?

Both are perfectly fine. In German, it is very common to use the definite article (der, die, das) for body parts when it is obvious whose body part is hurting. Saying Mein Fuß tut weh is also correct and frequently used.

How would I say 'The feet hurt' if both feet are in pain?

You need to use the plural form of the noun, which is die Füße, and conjugate the verb for plural. The sentence becomes Die Füße tun weh.

How do I specify who is experiencing the pain, like 'My foot hurts me'?

You can add a dative pronoun to show who is affected. To say 'The foot hurts me', you insert mir (to me) into the sentence: Der Fuß tut mir weh.