You might think about visiting Pskov and Novgorod (these are not far from St.Petersburg). Besides, the St.Petersburg area (region) has a lot of smaller towns around there (it's possible to use St.Petersburg as a base to visit the surrounding places, but coming back to stay in St.Petersburg at night). The largest cities closest to Moscow are Nizhniy Novgorod and Kazan (they might be convenient to reach by train), Vladimir is between Moscow and Nizhniy Novgorod, and it has some great historic things. Cheboksary is a night by train from Moscow, it's not bad in the summer. 

One of the cities I visited last year was Krasnoyarsk with its forest sanctuary 'Stolbi' (that's a great place for outings in Siberia, this reserve is in the vicinity of the city, but Krasnoyarsk is 5 hours by air from Moscow). 

 It all depends on what you are interested in and on whether you are ready to travel long distances by air or train. You can even travel to the Russian Black Sea (Sochi, Anapa, etc.) and have a standard sun and beach holiday (which is probably not interesting, given you are from LA).