You can find any sort of drink you want, some of the traditional beverages include:
Kumiss - fermented mare's milk, up to 6% alcohol content - imagine tart lemonade, mixed with semi-sour milk
Kumyran Shubat- fermented camel's milk
Kvas - described as similar to root beer it can be bought in a bottle in a store, or by the cup from people with giant yellowish tanks of it on the street
Cheap alcoholic drinks can be found at every little corner shop called the astanovka. These places are open 24/7, just knock on their door if the shopkeeper is asleep. Kazakhstan's specialty is cognac, though stores still sell vodka cheaper than bottled water at times. However, some of these astanovka sometimes sell alcohol of dubious origin; for the sake of your stomach you may want to buy your beverage in a supermarket, although the price will definitely be higher.
Several brands of beer, of good quality and flavor, are made in Karaganda.
Juices, in cartons, are common and delicious, especially peach juice.