There are hundreds or thousands of the little corner store variety. They can pack a lot of stuff in a small space and some have a lot more stuff than you would imagine. If you prefer the western type grocery stores, there are several:
A few of them:
City Mart (2 locations that I know of; Iran Street and The Diary.) The one on the Diary is two floors. They sell groceries, cosmetics, cleaning products, school supplies, toys, household goods, electronics. They do not sell clothes.
Shumailah Hari (off of Haddah Street)
They sell all of the above that City Mart sells including clothes (including hijabs)
Huda Market (Shariah Zubayri)
Sells foods, toys, and household goods.
Happy Land (Shariah Rowaishan).
Basically sells the same thing as Shumaylah Hari. You can find more American style baby items and toys here.