Despite the fact that Kemerovo is quite a large city, the overall low level of people's income prevents significant development of restaurants and bars. Don't expect to find as many places to eat and drink as you might find in a simarlarly sized Western city (more than 500 000 people).
Also about food: food is plentiful enough but, especially for Americans, the portions may seem small. Be prepared to order what seems like extra and may end up being just enough - it is especially true in small cafes and canteens. On the subject of ordering in low-end restaurants, you should also be aware that a menu is more of an opening point for negotiations than a list of what's really available. So when you find something that looks good and point at it, be prepared to be told that it isn't available today, or the cook doesn't want to make it, or something like that. Of course if you speak Russian, or have a Russian-speaking friend or associate with you, that will make this process much easier.
Probably the best restaurants of the city are Ellinikon and Richard. Barge and Pint are also good.
Ellinikon is a restaurant and bar on the premises of Greek Cultural Centre. Food is exceptional, and the service is really good and fast, so you won't have to wait for your steak for half an hour. Average dinner is around 400 Roubles per person ($14.00), not including drinks. The bar is not expensive, but a bit limited in choice of drinks. Pint of Baltica beer will cost 35 Roubles. See the map
Richard is also very good restaurant with "pretension". It is located at the intersection of Lenin prospect and Chkalov Street. Food and drinks are very good, an average dinner is around 400 Roubles ($14.00), Bloody Mary is 80 Rubles. English menu is available. See the map
Barge is an Irish bar which speaks for itself. It is located at the intersection of Lenin Prospect and Michurin Street. You will easily notice it, there are always BMWs and Landcruisers parked near the place. Although the regular customers are "tough"' guys and their girlfriends, you are highly unlikely to encounter any problems with them, the atmosphere is not tense. Barge has two-language (Russian and English) menus, so foreigners will feel much comfortable here rather than in some other places. The average dinner is about 300 Rubles (about $10.00) plus drinks, Bloody Mary is around 50 Rubles (less than $2.00). Both food and drinks are of very good quality, unfortunately, the same cannot be said about the service - sometimes it'll take an eternity to wait for your order. Guinness is available. See the map.
Pint is the another Irish pub in town. The food is good, but nothing special. The service is not that good, and the wait seems fairly long for your orders to come. Prices are at Barge's level - dinner is 300 Roubles (10 dollars) not including drinks. See the map.
Also, if you're staying in hotel, you will definitely find a restaurant at the hotel.
Nord - nice little cafe in the middle of the city. Specializes in confectionery products, though also offers pizzas and alcohol. A lunch for one will be around 60 Rubles ($2.00). See the map
Corsair - despite calling itself a "cafe", it's more like a restaurant. It is very comfortable with nice food. Average dinner is about 300 Rubles per person ($5.00). See the map.
There are many of them in the city. Nearly every large enterprise or organisation has a canteen, where anybody, not just their employees can have a lunch. However, we would like to note a couple of very good places.
Canteen in GorProject - it is located on Soviet Square. Though relatively expensive for a canteen, 40-50 Roubles (around $1.60) for a good lunch, the food is of excellent quality, and also the choice is wide. See the map
Canteen in Arbiter Court - acceptable food for acceptable price, 30-40 Roubles per lunch (1,3 dollars). See the map
World Pizza - located at the intersection of Sovietskiy Prospect and Vesennyaya Street, right in the heart of the city. Average lunch will cost around 60 Rubles ($2.00). Alcohol and smoking are banned here, so it's a good place to come with children. It is around the corner and down the street from the Kuzbass Hotel. I was told by our interpreter that it was opened by an American entrepreneur. Pizza is sold by the slice and it can be very different then you might expect at home, but it is always good. See the map
Zolotoy Tsyplenok (Golden Chicken) - located on Vesennyaya street, near Kuzbass Hotel. Obviously from its name, it specializes in chicken, while other options are also available. Lunch will cost around 60 Rubles ($2.00). There is a no-smoking area, but they sell beer. See the map
Silver-Pizza - pizzeria network, one is located at intersection of Krasnoarmeyskaya and Kirova Street, another one is at Stroiteley Blvd. and one more is recently opened at Lenina Prospect. Though being part of the same network, those are different places. Those at Krasnoarmeyskaya and at Lenina have rooms for kids, so those are good places to come with your family. The other one, at Stroiteley is nearly full every evening with people drinking beer, so that's the last place you'll bring your family to. Average lunch is also around 60 Rubles ($2.00). See the map of Location 1, Location 2, Location 3.
Podorozhnik - network of small pavilions offering different sandwiches. You'll get fed for about 30 Rubles ($1.00).