Learn Italian: Lavo le forchette e i piatti. - I wash the forks and the plates.

No, in Italian you generally must repeat the word for "the" before every noun in a list. Since forchette (forks) is feminine plural and piatti (plates) is masculine plural, you need to use both le and i.

Does i piatti only mean "the plates"?

While un piatto means "a plate", the plural i piatti is also used as the general term for "the dishes". The phrase lavare i piatti is the everyday way to say "to do the dishes" in Italian.

Why isn't the word io (I) at the beginning of the sentence?

In Italian, the verb ending usually tells you who is doing the action, so subject pronouns are dropped. The -o ending in lavo already means "I wash". You would only add io to add strong emphasis, like saying "I am the one washing them."

How do the noun endings match the words for "the" (le and i)?