You should follow the same elementary safety rules in town as you do at home.
Don't go down deserted alleys on your own, in the day or at night. You also really should not go into the poorest or most dangerous districts unless you are in the company of a local. (Ask your hotel or some local friends which areas to avoid).
You should also avoid going into bars that have bad reputations or ones where you just don't feel at home.
The Russian mafia really exists, so be careful not to cause problems, especially with local women or with other, dubious looking, men.
Alcoholism is a real problem, both for public health and socially. It is not unusual to see drunks in the street, even in the middle of the day. Try not to get close to them and if they are in a group then change pavements.
If you are driving, be very careful not to succumb to road rage and calling the locals names - it can easily become a fist fight.
The main thing is to avoid causing any problems because if you have to deal with the local police you are likely to become even more annoyed and exasperated (I can vouch for this).
However, Russia is no more dangerous than any other country! You just have to obey certain basic rules. You wouldn't get involved in a fight at home, or wander off into rough areas of towns you don't know. It's just the same in Russia.
In the countryside, especially in the forest or isolated areas, never go anywhere without telling someone where you are going and when you are expected back, or without the necessary equipment for your expedition (satellite telephone, anti-bear spray, etc.) It's very easy to get lost in the taiga and it's likely that no one will find you for several days...or even longer.