Learn Italian: Compro tre fette di prosciutto.

Fette is the plural of fetta, which is a feminine noun. In Italian, feminine nouns that end in -a in the singular change their ending to -e in the plural.

Why do we use just di and not del before prosciutto?

When you express a specific quantity of a food or material, like a slice, a kilo, or a glass, you just use the simple preposition di without combining it with a definite article. That is why it is tre fette di prosciutto rather than tre fette del prosciutto.

Can compro mean I am buying as well as I buy?

Yes! In Italian, a simple present tense verb like compro translates to both I buy and I am buying in English. You often use this simple form even when you are doing the action right now.

Why is there no word for I at the beginning of the sentence?

In Italian, the ending of the verb tells you who is doing the action. The -o ending in compro already means I buy, so adding the pronoun io is not necessary unless you really want to emphasize that you are the one buying the ham.