Yes — The Testaments (2019), co-winner of the Booker Prize, is the direct sequel set fifteen years after the end of the first book. It is told from three perspectives, including Aunt Lydia, and follows events inside Gilead and its resistance. The two novels form a complete story.
No — the books are self-contained and predate the TV series (2017–). The Handmaid's Tale TV show follows the first novel but extends significantly beyond it in later seasons, with storylines Atwood had no hand in. Many readers prefer to read the book first for Atwood's precise prose and tightly controlled point of view.
The Handmaid's Tale is a first-person, claustrophobic narrative — one woman's experience inside Gilead. The Testaments has three narrators and a more plot-driven, thriller-like structure. It answers questions left open in the first book (how did Gilead fall? what happened to Offred?) and gives Aunt Lydia a depth that reframes everything. Tonally they are quite different.
The Handmaid's Tale is widely taught in high school and university curricula. The content is mature — it deals with sexual slavery, state-sanctioned rape, and totalitarianism — and is typically considered appropriate for ages 16 and up. Many schools assign it with contextual discussion around its political and historical themes.