I spent 2 months living in St. Petersburg for an intensive language program. In terms of places to stay, I would look for hotels on one of the islands, as Nevsky Prospekt is very expensive. I stayed on Vasileevsky Ostrov. It takes about 10 minutes by Mashrootka to get across the river. Once you are on the main island - travel through metro is very easy if you at least understand how to READ Russian. There are not any signs in English, nor do the loudspeakers over the metro announce in English. Also - you are not permitted to take pictures in teh metro stations. The bridges to the islands close at midnight (maybe 1) for freight to pass through, then open again at 5 am. Unless you are a party animal...make sure you are on the island you want to be on.
There are some amazing restaurants - a great sushi chain with a buy one get one free happy hour - including beer, I do not remember the name, but they are all over and have a big Bonzi tree behind the name. There are also some amazing Georgian restaurants. Stop into foreign food restaurants - they are all awesome. Il Patio, on Nevsky, has a great Happy Hour as well. St. Peter and Paul's Cathedral is a must - you can see all the Romanov graves. I would also recommend you see Peterhof. You can jump on a boat right behind the Hermitage and spend the day. If you have a student ID...bring it everywhere. You can get cheaper prices, often, if you speak the language well enough, they will let you slide. If you research your Mashrootka/Bus routes, I would definitely recommend going to Tsartkoye Selo...it is about a 30 minute Mashrootka ride.
When you walk along Nevsky - take a stroll down some of the side roads - there are hidden treasures and restaurants everywhere. If you plan on going to bars, go to the Red Lion, it is just along the Neva on the main Island. They have ladies' nights on Thursday - free champagne for women all night.
If you are a Western man - be very careful around the police. They are very corrupt and will ask you for your documents. If you do not have them, they will threaten to take you to jail if you do not produce 500 rubles. Also, be very aware of the Russian woman around you - they love Western men
Women are very well respected, I did not have ANY issues while I was there, but several of my male, Western counterparts did.
Travelling to Moscow is easiest by overnnight train. I believe the train tickets are at Moskovskaya station, if not, they are at Mayakovskaya station. There are always really long lines and they close the ticket windows at lunch, for breaks, and at close - no matter how many people are in line. If you are not familiar with train travel in Russia - I would get the 1st class cabins. The standard tickets put you in metal cots with 4 bunks...it is freezing. The traffic in St. Petersburg and Moscow is horrible. The metros are very convenient - no need for a car. You can also put your hand out and a local will pick you up in their car. You arrange a price before getting in and give them the address. This is NOT recommended if you do not speak Russian well, are travelling alone, or are intoxicated.
Bring change or wipes - they charge you for toilet paper - per square...seriously.
If you have a dual entry visa - I would suggest a side trip to Finland. It is just across the Baltic. I went to Novgorod - you can pick up a bus in front of the shopping mall, there are also several different bus tours you can buy there. They leave as early as 7:00 AM.
I hope this helps. If you have any questions, please let me know, I would be happy to help.