I would not recommend taking buses in Russia. Your best bet is a high speed Sapsan train (multiple daytime and evening departures, takes about 3 hours 40 minutes) or a sleeper train (slower, overnight. e.g. the Red Arrow train, takes about 8 hours). You can buy tickets on the Russian Railways website (it quite horrible but it works).

Many will probably disagree but I think Moscow is not an interesting city, give it 2 days and St Petersburg is very interesting and much more pleasant, 3-5 days should be enough to cover both the city itself and the country palaces and museums around it