Learn Italian: Le mie scarpe sono nuove.

In Italian, you almost always need a definite article (the words for the, like il, la, i, or le) before a possessive word. Because scarpe (shoes) is feminine plural, you use the feminine plural article le together with the feminine plural possessive mie.

Would I ever use i miei for my shoes if I am a man?

No, because possessives in Italian match the thing being owned, not the owner. Even if you are a man, the word scarpe is feminine plural, so it will always take the feminine plural form le mie scarpe.

Why does the word for new (nuove) end in an -e here?

Descriptive adjectives have to match the noun they describe in both gender and amount (singular or plural). Since scarpe is feminine plural, the default adjective nuovo (new) changes to its feminine plural form, nuove.

I learned that sono means I am. Why is it used here for my shoes?

The verb essere (to be) uses the word sono for both I am (io sono) and they are (loro sono). Here, you are talking about your shoes (they), so sono translates to are.