Avion, Apark and Polus City center are the three major shoping centers in Bratislava but there all horible unless you whant nothing but overpriced coffe and high fation shops. If you have seen any shoping center in the world you will know what to exspect.
My advice is catch the 9 or 16 tram away from the city center untill you reach Zarhanicka. You will find Militichova maeket, grate slovak traditonal food, best Langosh in town. Althoguh if your aim is to buy quality cloths go to Aupark the cloths at Militichova are are all cheep copys and nock offs.