Being on vacation in Russia, you have to follow the same safety rules as in any other city of the world. Russia is not more dangerous than European or American countries. Of course, there is a street crime in Moscow, but probably, not as much as in your own capital city. Don't take all money with you when you are walking in the streets. Don’t put all your money in the same place, especially, if you are going to crowded places such as subway or touristic places. It is better to exchange currency inside the banks not in the exchange spots in the streets. Don’t show a large amount of money it can attract pickpockets. It is better to avoid walking in the dark streets at nights.
You always have to have your documents (passport with visa and registration) with you when you on the streets in Moscow or St. Petersburg. In case you have been stopped by policeman you have to show it to him.
Be careful on the roads too. Russians drivers have no respect for pedestrians, although this is not so true outside the big cities. If you travel by car it is better to leave it in special secured sites.
Generally, Russia is the same as any European country in terms of health risks. So, if you’re OK living in Europe, US, Canada, Australia or elsewhere you don’t need to take any special preventive measures before coming to Russia.
Take care of yourself and enjoy your trip to Russia!