Learn Hardcore French: Nous achetons une variété de légumes frais au marché. - We buy a variety of fresh vegetables at the market.

Unlike English, French does not allow dropping the subject pronoun. Every finite verb must have its explicit subject (je, tu, il/elle, nous, etc.). Here nous means “we” and must appear directly before the verb achetons to show who is doing the action.

Why is achetons spelled without an accent grave on the first e, whereas I’ve seen j’achète with an accent?

In the present-tense conjugation of the verb acheter, the stem-changing accent (è) appears in all singular forms and in ils/elles, but not in nous and vous. So you get:

j’achète, tu achètes, il/elle achète
nous achetons, vous achetez
ils/elles achètent
This keeps the pronunciation consistent (the unaccented e in achetons sounds like /ə/).

Why is the phrase une variété de légumes frais and not une variété des légumes frais or une variété de les légumes frais?

After expressions of quantity or measure (une quantité, une variété, beaucoup, etc.), French uses de (without an article) before the noun. So une variété de légumes means “a variety of vegetables.”
Using des (de + les) would turn légumes into “the vegetables,” which changes the nuance.

Why does the adjective frais come after légumes, and how does it agree?

Most French adjectives follow the noun they modify. Here frais (fresh) goes after légumes.
Agreement: légumes is masculine plural, and frais in masculine plural looks the same as masculine singular (frais). If it were feminine, you’d see fraîches.

What does au marché mean, and why is it au instead of à le or en marché?

au is the contraction of à + le, meaning “to/at the.” French requires contracting à + le into au.

à le marché is never used; you always say au marché.
en marché would literally mean “in market,” which isn’t idiomatic for “at the market.”

Could I simply say Nous achetons des légumes frais au marché? What’s the nuance?

Yes.

Nous achetons des légumes frais au marché. = “We buy (some) fresh vegetables at the market.”
This is perfectly correct and more general. It doesn’t emphasize “a variety” (diversity) of vegetables, just that you’re buying fresh vegetables.

What extra meaning does une variété add that des légumes frais alone doesn’t convey?

une variété highlights diversity or a selection of different types.

des légumes frais = fresh vegetables in general.
une variété de légumes frais = a variety of fresh vegetables, implying multiple kinds or a broad selection.