To keep it short: yes, sure! It is more than safe to travel to Russia! Many people have the feeling that Russia may not be a safe country to travel to, but they cannot be wronger. Travelling in Moscow, St. Petesburg and pretty much all Russia is more than OK on terms of safety.
Despite that, anywhere you travel (not only in your trip here, but everywhere in our huge world), try to reduce the small odds of something unexpected happening. It’s usually safe around, but Moscow is a big city, you know! You never be 100% sure.
It is always a good idea to look for your belongings, and be aware of where you go. In St. Petersburg, Moscow and other cities there are pickpockets in the subway and in some touristic places. Nothing to be really worried, but just to be aware of.
Another good idea is just to keep your wallet in your front pocket of your trousers, this is an easy thing to do but it can prevent someone to sneak behind and get it without you noticing.
Also, try not to look ridiculously tourist – again, this is a tip for your life, not only when travelling in Russia. Keep your camera in your bag when not using it, not hanging on your neck, for example. People getting too close behind are something to be aware of, too.
And in general, the golden tip is: stay in a place close to the city center, the Red Square and surroundings. These are the safest places in town, where you will least have to worry about any of this.
The Red Square are is just the best place to be in Moscow, with many good restaurants, parks, sightseeing and cultural activities going on. Safari Hostel is located at a short walking distance from it, and is a warm place for your stay in the city!