Some advice on getting around and first things you'll need in Russia.
Here are major transport points in the city as well as key locations.
Transport
Transport details for the airport: https://pulkovoairport.ru/en/transport/bus/
Travel card (pre-paid balance): http://podorozhnik.spb.ru/en/
To pay the fare, you should tap at the turnstile in metro or at the handrail in the ground transport. When you tap, the fare is deducted from you balance. In ground transportation there is staff selling tickets (if you don't have a card) and also checking whether you've tapped your card. (For this check, just present your card - the fee will not be deducted for the second time). It is cheaper to use card than to buy tickets.
You can buy and top-up travel card at metro stations (at ticket booth or machine). Mention "electronic wallet" ("elektronnyi koshelyok" / "электронный кошелёк" in Russian) to use this pre-paid balance option - otherwise you can buy other types of tickets, but they are more complicated to choose and use.
Money
Cards are accepted in many (most) places.
You may need cash to pay for your first bus ride from the airport. (Buses from the airport are mostly equipped with machines where you may pay by tapping VISA Pay-pass / MasterCard Pay-wave card, but there is a small chance that this option fails...).
At the airport if you need to exchange money there is a bank (check here: https://pulkovoairport.ru/en/map/). In the city also look for a bank rather than exchange office.
Taxi
Uber operates in St Petersburg along with local taxi-app services, Yandex.Taxi being most popular.
At the airport do not use the services offered by individuals - if needed, order a taxi through the special booth.
Mobile network
Mobile internet and calls are quite cheap and have relatively good coverage. To get a local SIM-card, visit one of the mobile operator's offices (MegaFon, MTS, Tele2, Beeline etc.). You will need a passport. Be attentive with tariffs and extra options to avoid "strange" costs.
We also use mobile internet at the White Sea, but this is another region and SIM-card from St Petersburg will be in roaming at the White Sea. A shared mobile modem providing wi-fi will be in our classroom for general use, but please avoid heavy traffic like video, audio, large images etc.