Learn Russian: Нам нужно нарезать овощей.

The prefix на- in нарезать indicates accumulating a quantity of something through an action. When a verb has this "accumulative" meaning, its direct object takes the Genitive case instead of the Accusative. This is what gives the sentence the meaning of slicing "a bunch of" or "a quantity of" vegetables.

Does the prefix на- here just make the verb perfective, or does it change the meaning?

It does both. While нарезать is often used as the standard perfective pair for резать (to cut/slice), the на- prefix inherently carries the nuance of producing a quantity of something. So while you are completing the action (perfective), you are specifically completing it by piling up a bunch of sliced vegetables.

Why does the genitive plural of овощ end in -ей instead of -ов?

The noun овощ (vegetable) is a masculine noun ending in the husher consonant щ. Masculine nouns ending in ж, ч, ш, or щ always take the -ей ending in the Genitive plural, making it овощей.

Just to review, why is the subject 'we' in the Dative case (Нам) instead of Nominative (Мы)?