There are a great many things to recommend eating and drinking in Srinagar. This is a predominantly Islamic and Hindu city, mind you, so it might be best to go easy on the drinking if you interpret that to involve some form of alcohol. While there are in fact some liquor places here as well as off-licences, they are generally located in the hotels and are Western-style. Beer can definitely be found, but it is best to consume in moderation. If your concern is eating, though, you can go ahead and indulge in the best food that the Kashmir has to offer without any guilt. Srinagar has some wonderful dishes for which it is known, as well as a willingness to pair Indian spices and the Kashmir flavours with dishes originating in other countries, specifically ones of Continental and Chinese origin.
Kashmir cuisine is predominant, of course. Shervani Road would be a good place to start for it. Lal Chowk, one of the best places to go shopping in the city, is also one of the best sites for eating and drinking in Srinagar. Juniper, a rather nice restaurant serving traditional local cuisine, is located here, as a matter of fact. Lal Chowk is also a great place to shop for superb pastries and other bakery products in the city. Most tourist attractions in the area tend to act like magnets too for good restaurants: for instance, Dal Lake sees quite a number of good restaurants, especially vegetarian ones and ones serving the famous grilled mutton of Srinagar. Mutton figures highly in Srinagar meals. Other proteins commonly used proteins would be fish and chicken. Indeed, it is not uncommon to find dishes that use both these proteins and mutton.
If you happen to be a vegetarian, do not worry: Srinagar cuisine is popular for its usage of vegetables. Look for delicacies such as kheer (rice pudding, also the offering made to the goddess Bhawani that resulted in her shrine nearby being called the Kheer Bhawani) as well as the rather deliciously savoury dum aloo (roasted potatoes in gravy).
Now if you want to try some of the drinks that the city has to offer without going the alcohol-laced route, look to the delicious teas that are brewed in the city. Kawah or kahwah is something that people eating and drinking in Srinagar absolutely have to try. This is a superb infusion that often has a green tea base (although versions exist that do not have green tea) and is spiced with some of the best produce of Srinagar: genuine and wildly aromatic saffron, sweet cinnamon (look for places that use real cinnamon and not cassia if you want the best flavour), the lovely almonds of the city, and some cardamom. Pink tea, another infusion popular in the city, has cardamom too but trades green tea for black, giving it a rather different flavour profile. It is typical to find this infusion blended with milk for a sweet, addictive taste that many tourists admit to enjoy.