No matter what time of year you visit, bring a sweater. St. Petersburg especially can be unexpectedly cold in summer. A raincoat and fold-up umbrella are also musts. You'll probably be doing a lot of walking outdoors, so bring warm, comfortable clothing, and be sure to pack a pair of sturdy walking shoes.
Russians favor fashion over variety in their wardrobes, and it's perfectly acceptable to wear the same outfit several days in a row. Be sure to pack one outfit for dress-up occasions, such as the theater. The layer system works well in the unpredictable weather of fall and spring; wear a light coat with a sweater that you can put on and take off as the weather changes. In winter, bring heavy sweaters, warm boots, a wool hat, a scarf and mittens, and a heavy coat. Woolen tights or long underwear are essential during the coldest months. Russian central heating can be overly efficient, so again, use the layer system to avoid sweltering in an overheated building or train.
Russian pharmacies, supermarkets, and hotels all have reasonable stocks of the essential toiletries and personal hygiene products, but bring your own supplies of medicines and prescription drugs you take regularly. Although some well-known Western brands are easily available, you may not recognize the Russian equivalent of certain medicines. Consider whether you might want to bring any items that can be difficult to find in Russia, such as insect repellent (in summer and fall mosquitoes can be a serious problem), camera batteries, laxatives, anti-diarrhea pills, travel-sickness medicine, and the like.
Toilet paper is plentiful in hotels but less so in public buildings, so bring small packages of tissues to carry around with you. Premoistened cleansing tissues will also come in handy, especially if you're traveling by train. A small flashlight may also prove useful, particularly if visiting someone's apartment—stairwells are often dimly lit.
Within Russia, the rules regulating carry-on luggage are strict disregarded. Checked luggage is frequently lost and/or pilfered, so pack as much as you can in your carry-on, including all of your valuables, for internal flights.