The quickest are the daytime high-speed Aurora trains, but most people actually take an overnight express. The Red Arrow/Krasnaya Strela is probably best known but there are plenty of others at varying prices. There are no real issues with safety on overnight trains in Russia (so long as common sense is exercised). These links should provide all the info you need:
http://www.waytorussia.net/Transport...tic/Train.html
http://www.lonelyplanet.com/thorntre...art=0&tstart=0
http://www.poezda.net/en/index
http://www.eng.rzd.ru/wps/portal/rzdeng?STRUCTURE_ID=9