If I enter Russia by taking a no-visa-needed Helsinki-St Petersburg ferry, can I take a day trip to Moscow, or do I have to stay in St Petersburg for my entire stay?
Here's the actual legal basis for visa-free travel: http://base.garant.ru/195505 . It says you can stay for 72 hours, have to leave by the same port where you entered, and most importantly (point #6) it says you may only be located on the territory declared in the "official tourist program" filed by your cruise line with the Russian government.
There may be some "official tourist programs" that include other cities like Moscow (see e.g. the article at http://www.rostourunion.ru/ob_rst/novosti/news_archive/pervyyi_bezvizovyyi_parom.html )
Some articles online imply that this requirement is just a formality. For example, this article ( http://www.myestonia.ru/publ/72_chasa_bezvizovogo_schastja/13-1-0-572 ) implies you can go to other cities as long as you still pay for your St. Petersburg "official" hotel. However, the legality of this is very questionable. I would strongly recommend getting an actual Russian visa if you want to do any real exploring, especially given the current tense political situation.