Arnhem's main shopping streets are the Ketelstraat and its continuation the Vijzelstraat, which together form the commercial spine of the centre. You will find the usual Dutch chains alongside more interesting independent shops.
For something different, head to the Weverstraat (off the Jansstraat). Three shops next to each other cater to different tastes: Deal for second-hand clothing, Britain for hip-hop and streetwear, and Soul for curated vintage from the 1970s.
The Bijenkorf department store is the biggest in town -- expensive, but the basement "Chill Out" zone has good gifts and trendy clothing. Vroom & Dreesmann on the Velperplein is the more everyday alternative.
On Saturdays, the market on the Marktplein is worth visiting. Walk down the Koningsstraat to reach it, and stop at Dille & Kamille in the same street for kitchen and garden supplies.
If you are hungry while shopping, Tsjakka in the Roggestraat does cheap and cheerful Dutch snacks -- try a kroket.