When talking winter sports, the oldest and most traditional pastime would be cross country skiing — though it’s more of a recreation activity than a sport. It is relatively easy and cheap (as opposed to snowboarding; you don’t need special wear and you don’t pay for lifts), and you can do it almost anywhere — just grab your skis, skiing boots, and go to some park. Of course, leaving the city is a lot better — this way, cross country skiing is half sport, half nature appreciation; also, it's a great way to spend time with your friends while doing some exercise (skiing through the winter woods alone might be a thrilling experience, but I wouldn’t recommend it to anyone). Also, you can bring some food and drinks, make a campfire, and so on — that’s also a great pastime with a whole tradition to it.
You can ski in several places in Saint Petersburg; still, if you’d like to follow my advice and get out of the city, the best destinations will be to the north — places like Kavgolovo, Pargolovo, etc (to get there, you should take a train from the St. Petersburg-Finlyandsky railroad station). Note that ITMO’s KronBars sports club has a skiing section, so giving it a try or at least asking for directions might be a great idea, as well.
A more "active" winter sport would be alpine skiing — which includes snowboarding, as well. Essentially, these two are the same, so I’ll refer to it as snowboarding, as it is considered to be the choice of the younger.
Snowboarding has been becoming more and more popular in Russia; as of now, there are many places where you can do it. One of the closest to Saint Petersburg is the Northern Slope, only 20 km from the city — it is great for beginners and it takes only about an hour to get there. Another good option for beginners would be Tyytari Park to the south of the city, as it has a special training zone which costs 350−500 rubles per day. Yet, if you are looking for something more advanced, you can try KrasnoyeOzero (Red Lake) or Igora— though more expensive, it is one of the most popular ski resorts around Saint Petersburg. Overall, prices for lifts range from 50−150 rubles per lift or 400−800 per hour, renting equipment will cost around 800 an hour. You can find more information on the Internet, here, for example — unfortunately, it’ll all be in Russian so you better consult your Russian peers. And, as with cross country skiing, there’s an alpine skiing section at the KronBars sports club, as well.
If learning something new is not what you’re after, and you just want to experience the fun of riding down the slopes, there’s tubing — in Russia, we call the inner tubes you ride "vatrushka's", as they resemble this traditional Russian pie. Tubing is most fun and doesn’t imply having any skills; you enjoy the rides, you hurl yourself into snow — well, I just can’t imagine a person who wouldn’t like it. Tubing is available at most slopes and ski resorts.
Ice-skating is yet another winter activity you can do in Saint Petersburg. It is a great way to spend an evening, even if you don’t really know how to skate — learning the basics is easy, and tripping is just a fun part of it. One of the greatest advantages of ice-skating is that you don’t have to go somewhere far to do it, as there are ice rinks in almost every district of Saint Petersburg, and if you’re tired, you can always just call it a day and go home. The biggest rinks close to the city center would be the ones in the Yubileyniy sports complex (Sportivnaya subway station), the Ice Palace arena (Prospekt Bolshevikov), the Kirov Central Park on Elagin Island and near the Young Spectator's Theatre (Pushkinskaya subway station). Prices range from 150 to 500 rubles per hour, and about 200−400 rubles per hour for renting the skates.
If you are looking for something really original — there’s ice fishing.
Personally, I can’t call it a sport or even an activity — for me it’s more of a staring contest. In essence, ice fishing is fishing through an opening in ice on a frozen body of water, and is about patience and … love of fishing. As strange as it may seem, there are many fans of this activity, who find it entertaining, even addictive — so who knows, maybe you’ll become one of them. Still, you should bear in mind that ice fishing can be considerably dangerous (you may fall through the ice into freezing water, and it’s really hard to get out as ice breaks when an inexperienced person tries it), so if you’ve decided to try, you’d better do your homework first. Also, it is obviously best to first try ice fishing at some recreation center near Saint Petersburg, as there they can provide you with the necessary equipment and show you the fishing spots.
As you can see, there is a lot you can do in Saint Petersburg with regard to winter sports — just remember to plan everything carefully and keep yourself warm. Winters can be really fun in Saint Petersburg — so getting ill is not an option!