In general, it is very cheap to dine in Kiev by European & US standards. So long as you stay away from the places that totally pander to tourists, the food is great and cheap. Try the Borscht and the Mlyntzi and then try absolutely everything else. Baked goods are cheap and great too. Even the ice-cream on the street is great. An especially distinctive one is to the right from Khreshchatyk metro exit - blue kiosk with varying lengh of queue.
When you see vendors selling some liquid from big yellow/blue tanks on the street, you can be sure that it is "Kvas," which is a brewed bread drink. Some people like it and others hate it. It tastes a bit like malt, and the alcohol content is so low (0.05-1.44%) that it is considered acceptable for consumption by children. Try "Odyn Malenkyi" (one small) drink.
You should not drink the tap water. It is advisable to buy 5l. bottles in the supermarkets; they usually have English section for "ingredients". You can always order "Bonaqua" (sparkling mineral water), but beer is just about as cheap.