Of course considering an area size of Russia, it is difficult to give any sort of general advice about the climate and weather except that summers are warm to hot, and winters get very cold in some areas. In general, the climate of Russia can be described as highly continental influenced climate with warm to hot dry summers and (very) cold winters with temperatures of -30°C and lower and sometimes heavy snowfall.
The weather in Moscow, St. Petersburg and Volgograd varies greatly according to season. The winters are long and cold (with lots of snow and average temperatures around -10 degrees Celsius), and the summers are hot with occasional thunderstorms. As for the types of clothes to bring, it depends on the season of the year, but remember the two crucial points about life in these cities: 1) You will be walking a lot; feet and public transportation are the main modes of transportation in large Russian cities, so your shoes will take a beating and will get quite dirty, i.e. light-coloured shoes are not recommended 2) The same goes for clothes. Russians do wear bright things (shoes included), but dirt naturally stands out more on brighter clothes than it does on darker ones. So, you can see why a tendency toward wearing darker clothes can be seen on the streets of Russian cities. No one wants to be washing their clothes every day! For more information on the weather, try visiting CNN Online or the BBC website. If your Russian is good enough, you can find the current broadcast here.
Northern & Central European Russia has the most varied climate; the mildest areas are along the Baltic coast. The Summer is mostly dry and sunny, sunshine can be observed during nine hours a day, with local showers or thunderstorms in the afternoon and evening. As almost everywhere in Russia, winters can be very cold with frost and snowfall, the first Snow often falls in early October. Spring and Autumn can be quite unsettled, sometimes low pressure systems can bring frequent rain- or snowfall and strong winds.
Southern European Russia: Winter is shorter than in the north. The Russian Steppes (in the southeast) have hot, dry summers and very cold winters. The north and northeastern areas around the Black Sea have milder winters, but frequent rainfall all the year round. The climate in this region is slightly mediterranean.
Siberia: This area of Russia is famous for its extreme climate with very cold winters, but warm to hot summers, although they tend to be short and wet. Average Precipitation is about 500 mm, most falls in the form of rain in spring and Autumn. The winter is mostly dry, snow covers the ground from end october to mid march in some years. In the winter the weather can be very dangerous sometimes with the temperatures below -35°C, strong winds and snowstorms. The coldest place in Russia is Oimjakon in the central Sibirian region Jakutia, where the winter temperatures are almost the lowest in the world, below -50°C in January. The Artic sea is Ice-bound from November to the end of March.