You are right, in Dubai there are a lot of place you can buy stuff.

Good quality and normal price stuff, visit following place.

Toys r us,
Mother Care
Baby Shop

For clothes you can buy them at many different shops and malls, do visit, mall of emirates, city center, Outlet mall (have some good stores), Gap, Marks & Spencer, and the list goes on & on,

i still prefer mother care because they stay good even after wash, also the same for Marks & Spencer.

Good Luck, do let me know if u need any further help

regards