Providence's most interesting shopping is not in malls.
Thayer Street on the East Side (near Brown) has a mix of independent bookshops, vintage clothing, and student-oriented shops. It's walkable and worth browsing.
Westminster Street downtown has seen independent retail move in alongside the converted Arcade building. The stretch between Weybosset and Mathewson has design-focused shops and a few galleries.
Federal Hill is the place for Italian specialty food: imported pasta, cheese, cured meats, and baked goods at the delis along Atwells Avenue. Venda Ravioli at 265 Atwells is worth a stop for imported grocery items and freshly made pasta.
Providence Place Mall (one Francis Street) handles the standard national retailers if you need them — large format, easy parking, right off I-95.