Is bamboo a vegetable or a grass?

Both, according to http://www.britannica.com/plant/bamboo.

It's a vegetable as bamboo shoots in a Chinese takeaway, and it's grass when a panda or gorilla is eating it or when it's it's just growing somewhere wild. https://en.wikipedia.org/wiki/Bamboo

The question has little meaning because 'vegetable' and 'grass' are names from different ways of classifying things. Some forms of bamboo are a vegetable (if you mean a plant that can be eaten) but most are not.
All types of bamboo are in the botanical family Gramineae (which are commonly called grasses) but it is not a grass in the commonly accepted meaning of the word (a low growing monocot that can be cut or eaten and reshoot easily).