Learn French: Il achète des lunettes et un chapeau.

Most French nouns take an -s in the plural, but words ending in -eau are irregular and take an -x instead. If he were buying multiple hats, it would become des chapeaux.

Why is there an accent on achète if the infinitive is acheter?

Acheter (to buy) is a stem-changing verb. To make it easier to pronounce, French adds a grave accent to that e for all forms except nous and vous. So it becomes j'achète, tu achètes, and il achète, but stays nous achetons.

Can I use the singular une lunette for a pair of glasses?

No, just like in English, glasses for your eyes are always plural in French. You must use des lunettes or une paire de lunettes. The singular une lunette usually refers to a telescope.

Should I make a pronunciation link (a liaison) between et and un?

No, you must never make a liaison after the word et (and). You just pronounce them separately: the t in et is always silent.