If you’re going on the trans-siberian/manchurian/mongolian, get yourself a copy of trans siberian trailblazer by Bryn Thomas - it’s got guides to St Petersburg, Moscow and the other cities, but the best thing about it is the guide to the railway itself, which includes things to look out for at specific km points on the railway itself (there are markers at the side of the railway), it was great knowing what to look out for when you’re on a 5 hour stretch (loved the trip, helped having an irishman in the cabin who i could chat to, but just a lovely rotation of snoozing, reading, looking out the window, listening to music, chatting, snoozing, etc. Always feeling like you’re getting closer to your destination). I was in Moscow in late January, it wasn’t really a problem for a tourist (apparently, it was the coldest Moscow had been since World War 2 when I was there, -40. They called it the baptising frost, although that might be every year when it gets super fucking cold. I made an icicle on the end of my runny nose)
As said, the Moscow metro is well worth a tour, avoid the police as mentioned, look out the bistro places for cheaper food (often just off main streets). I stayed in a hostel called Sweet Moscow, but I think that’s probably shut now (2006). I remember more specifically about St P, but Moscow was great to just wander around.