Learn Polish: Gdzie trzymasz cukier? - Where do you keep the sugar?

Cukier is a masculine inanimate noun. For these nouns, the accusative case (the form used for direct objects) is exactly the same as the basic dictionary form. So, it stays cukier.

I learned that trzymać means "to hold". Is it also used for "to keep"?

Yes! While trzymać can mean physically holding something in your hands, it is also the natural verb to use for storing or keeping items in a specific place in your home.

Could I say Gdzie masz cukier? instead?

Absolutely. Gdzie masz cukier? (Where do you have the sugar?) is very common and means almost exactly the same thing. Trzymasz just slightly emphasizes the idea of storing it.

What happens to cukier if I make the sentence negative, like "I don't keep sugar here"?

In negative sentences, the direct object changes to the genitive case. The genitive form is cukru (e.g., Nie trzymam tu cukru). Notice that the e disappears when the ending is added, which is a very common pattern in Polish known as a "fleeting e".