Moscow is a safe place. Like any big city however, basic precautions have to be taken whether you are visiting short-term or staying in Moscow for a while. Below precautions are valid not only for Moscow but most large cities in Russia.
Never exchange money in the airport. Not exactly a safety threat, but exchange rates in airports especially before customs is a rip-off. You will get almost double Rubles for your currency exchanging it in the city center. Use ATM to withdraw money or pay by card, cards are taken almost everywhere.
Do not use taxi drivers offering their service at the airport arrival area. When you arrive to any Moscow airport, upon exiting from customs area you will be met by a crowd of taxi drivers offering rides to Moscow. Never use any of those drivers, no matter how official their badges look. Use express trains. If express trains do not work for you, you can order taxi from official taxi counter (available in all airports) or through apps like Yandex taxi.
Do not take taxi on streets. Use apps like Yandex taxi or Gett (acquired by Yandex). If your business contracts taxi company use their service even if there is wait to get the car.
Do not buy sim cards from street sellers. Go to an official retailer. You will need passport to buy a card.
Beware of pickpocketing, especially in public transport. Keep your passport, wallet and other valuables secure. In subway don’t stand near the doorway playing or texting on your smartphone.
Do not leave your luggage, bags, valuables out of sight even for a split second — everywhere.
Be extremely cautious crossing streets. Even if you are on pedestrian crossing, going on green light, be absolutely positively sure that cars actually stopped before the crossing.
Never give money to any beggars, especially in subway.
Always check bill, when eating out, before paying, to make sure you pay for what you ordered. Information on related subject — tips, can be found here.
Avoid walking in remote areas of Moscow or have company (preferably a local).
Do not walk alone in the dark, in areas where there is no one around, even in the city center.
Do not talk to strangers, asking for your attention on the street.
Do not eat shawarma. Exception can be made to shawarma in cafes and restaurants with table service.
In winter — do not walk close to buildings due to danger of snow and ice falling from roofs. Wear proper winter shoes, as it gets very slippery in Moscow in winter season. Especially be careful on underground passage stairs, as they are made of polished stone and get slippery even in rain. Some guidance on clothing in winter can be found here.
In summer — beware of ticks. Here is separate article on the subject.
Wash hands as often as you can. Have a pack of wet napkins or anti-bacterial fluid with you.
Do not do your own driving in Moscow, unless you are fully and mindfully confident about driving in Russia.
Do not pay in any currency other than Rubles. This is illegal and there is risk of a fraud.
Beware of scam and fraud of all types still high in Russian business and private dealings. Be extremely cautious buying products and services appearing to be very cheap. Russian wisdom says: “Stingy person pays twice.”
If you are in trouble and there is no one local to help you, look for police with sticker on their uniform saying — “Tourist police”. Their job is to assist foreigners and they (well, in theory) speak English.