I'm sure shopping in Dubai can be quite daunting for newcomers. The city has really reached the pinnacle of the shopping experience. While the guide to shopping malls in Dubai should be your first stop, here are a few brief suggestions:
The gargantuan Dubai Mall is right up there with the biggest shopping malls on the planet.
Ibn Battuta has gorgeous interiors, inspired by its Moroccan explorer namesake, and is home to many recognised retail brands such as Marks & Spencer and Decathlon.
Mall of Emirates is truly astonishing as it has an indoor ski resort. If you get tired of the slopes, you can enjy a toboggan ride or playh with penguins.
Check out the Dubai guide on Expat Arrivals for a more comprehensive overview!
Additionally, Al Safa Centre is a small shopping centre in your area where I'm sure you can get some of the necessities.