It is October 31st and Moscow is hit by the first snowfall, although this weather is not usual for October - November. Normally we have such snow somewhere end of November or most likely in December. If we consider this snow as winter beginning, this means winter in Moscow is five months long (March is a full scale winter). This snow is going to melt anyway, because weather in November and December is quite unstable. Temperature fluctuates from slightly freezing at night to above zero during the day.
If you are coming from northern hemisphere you are probably very familiar with how to cope with cold and snow. Canada, northern US states (Montana, North Dakota, New England, Wisconsin and neighboring states); Scandinavia have winter weather conditions close to those in Moscow. I would say for Moscow you chose same clothing style as you would in one of those areas. For those coming from places with milder climate , below are suggestions on clothing in late fall and beginning of Moscow winter.
Hat. It can be something simple, yet practical. "Classic" fur hats, so popular in soviet times no one wears anymore. Beanies are popular here. Russian men prefer conservative style and colors that is black and gray. Women on contrary normally pick fancy hat styles.
Waterproof, warm garment preferably with hood. You can use umbrella instead of hood, which however can be inconvenient. I emphasize waterproof because many locals wear wool coats and fur coats and they just look cool, but not very practical in the beginning of winter. In November in December it is very usual to have rain, sleet or snow, and type of precipitation can change during the day. In the past years we had a number of occasions of freezing rain.
Extra sweater or fleece jacket or warm piece of clothing beneath upper garment. This is optional, to add extra layer of insulation. Usually all offices, apartments and public places have really good heating inside, but some place may have draughts and better to have something warm on, even inside.
Gloves. Whatever you like. Most people wear leather gloves, but wool gloves would do just fine.
Shoes or boots on thick sole, waterproof and preferably with thick socks or some extra layer of insulation to keep your feet warm. This is especially important because in Moscow it snows all the time in winter. I mean it, there will be piles of snow by the end of January. Because of temperatures going up and down, and because of special chemicals used to melt snow, the snow turns into sludge. Thick layer of sludge on sidewalks, roads, everywhere. You are out of your place or office, put your feet onto the street and it fails into a mixture of snow and water up to your ankle. It is also very slippery, so choosing proper shoes is a safety measure. Speaking of those special chemicals to melt snow and ice on streets, generously applied by street workers on roads and sidewalks and everywhere. Those chemicals will ruin your shoes easily, if they are made of natural materials. Advice is not to wear expensive shoes (unless you are in a car and don't walk much). What many Russians do, they just have spare pair lighter shoes for the office and something heavier and more practical for the street.
Style. You will find variety of clothing styles on Moscow streets. Many people tend to dress conservatively in what they believe stylish in their own understanding. Younger generation is much more relaxed about outfits. Colors of choice are normally gray and black. Unless you have some specific reasons to dress in a certain way (like office policy), choose outfit that suits you best. I would say most common European and American outfits would look just fine in Moscow crowds.