Where to? đi đâu, no ở

When a verb of going is followed directly by the place, đâu takes the place's slot, with no ở: Anh đi đâu? Where are you going? The same goes for về (go back), tới (come to), lên and xuống (go up, go down).

The answer shows why. You say đi chợ (go to the market), with no word for "to", so the question has none either. Ở means "at": it's for where someone is or where something happens, as in Anh làm ở đâu? Where do you work? So when đi comes before another verb (đi học, đi làm, đi chơi), ở đâu asks where that happens: Em đi làm ở đâu? Where do you work?

Đi đâu vậy? is also an everyday greeting when you see someone heading out. It's meant as a friendly hello rather than a real question, and a short answer is enough: Đi chợ.

In quick speech, a thing followed by đâu asks where it has got to: Chìa khóa đâu rồi? Where have the keys gone? Điện thoại em đâu? Where's my phone? To ask a stranger where something is, ở đâu is the safe choice: Nhà vệ sinh ở đâu? Where's the restroom?

After không, đâu means "anywhere": Em không đi đâu. I'm not going anywhere.