Learn Italian: Vorrei dei pomodori, delle olive e degli spaghetti. - I would like some tomatoes, some olives, and some spaghetti.

The Italian partitive article (meaning "some") is formed by combining the preposition di with the definite plural articles. The form changes to match the noun that follows it. Pomodori is masculine plural starting with a standard consonant (di + i = dei). Olive is feminine plural (di + le = delle). Spaghetti is masculine plural starting with s plus a consonant (di + gli = degli).

Could I just leave out the words for "some" and say Vorrei pomodori, olive e spaghetti like we sometimes do in English?

Yes, but it sounds a bit abrupt in Italian. While you can drop the article to make a generic list, using the partitive articles (dei, delle, degli) is the most natural and standard way to express an unspecified but limited quantity of countable items.

Why are all the partitive articles here plural? Could I use a singular one like del?

You use the plural forms (dei, delle, degli) for countable nouns, like tomatoes, olives, and spaghetti. You would only use the singular forms (del, della, dell') in this sentence if you were asking for an uncountable mass noun, such as del formaggio (some cheese).