I'm traveling to Beijing, Xi An and Shanghai. How much does food cost? How much money should I have?

It depends on what kind of food do you plan eat in these places. It is hard to tell you the exact cost. When I was in Xian, I usually spent about 50RMB per day, because my friend usually took me to some local food restaurants.
Generally speaking, the Chinese food is cheaper than the Western food, while price in Xian is much cheaper than that in Shanghai and Beijing.
In my opinion, you do not need to take so much currency. Credit card can be used in these cities.