Appetizers: In a full course Russian meal, you would usually have both an appetizer course (with prepared salads and things like canapes) and a soup course. But ordering in a restaurant you'd probably only choose one. I adore Russian soups. My favorite hot soup is called Rassolnik. It is made with a very special type of preserved cucumber. These pickles are very salty, not as sour as ours. Another very good one (that also usually appeals to American palates, I've heard) is Solyanka. It means "salty." It has many kinds of meat in it and is very flavorful. Russian soups have a slight dimension of sourness to them, which I like. My absolute favorite soup is called Okroshka. It's a cold summertime soup made with the rye bread beverage kvass. I love it, but I also love kvass (a lot of people don't but a lot of people do). For salads, two popular and tasty ones are called "Vinegret" and "Olivier". These are cold salads made with cooked, finely chopped vegetables dressed with a mayonnaise/cream based dressing.
Main courses: Pelmeni are really good, they are basically Russian tortellini, but they're pretty much always filled with the same thing: a mix of ground pork and beef and finely minced onion. Golubtsy, or stuffed cabbage leaves, are also very tasty. Both those are filling and I like them even though the flavors are simple, they are good. Russians don't generally have the same idea of a main course that we do: a plate neatly divided into three piles, meat, starch and vegetable. From what I've seen, the main plate very commonly consists of a cooked meat/fish, and then there may be something with it, like rice, potatoes, or kasha which is cooked buckwheat groats. When I tried to make it it was too mushy, but when my boyfriend's mom cooked it it was one of the tastiest things I've had!
Desserts: I like simple blini with honey! Another good thing, this is more Ukrainian but they may have it in a restaurant, are vareniki with cherries. These are boiled dumplings with cherries inside. There are also something called piroshki (pirozhok is singular) that can be a dessert, it's a baked sweetish bread filled with jam or cottage cheese. (Piroshki can also be savory, and you may also see them on the appetizer menu. Another good choice for an appetizer).
Drinks: I don't know about any alcoholic drinks, but I really like kvass. Although I don't know if the one you order in the restaurant will be as nice as the one you buy from street vendors. Compote is also good, it is made by boiling fruits in water and sweetening with honey or sugar. The fruit is separated from the liquid, and the liquid is served as a beverage called 'compote'. Very good in the summer!
I hope it helps, but I don't know if it necessarily will, since my knowledge of Russian food doesn't come from trying it in a restaurant. Most of it is what my boyfriend's mom has made and taught me to cook. But, those are the dishes that I enjoyed in Ukraine, a few in restaurants, most in the home.
This site has some nice recipes: http://www.russlandjournal.de/en/recipes/
The good thing is there are also pictures, so maybe you can identify what you'd like to try.