You can find the branches of all well-known banks in major Russian cities - Barclays Bank, Raiffeisen, Citi Bank, etc. You can even open an account with these banks from your own country. Opening a bank account in Russia will take you a few minutes and you will receive the same service as at home.
When going to smaller cities in Russia, it is better to deal with a Russian bank, (VTB24 or Alfa Bank) which offer good service. Moscow International Bank offers a wide range of services and you can open USD, ruble and euro accounts. They have an English version of their online banking, most of the pages are translated (which is more than you can expect from most other Russian banks) and they also have English-speaking operators available at their service-line.
Types of bank accounts
There are different types of accounts you can open in Russian banks. The most common are current and savings accounts. Current accounts are more convenient for everyday payments.
In some cases it makes sense to have different types of accounts for different purposes. For example, a ruble account can be useful to receive your Russian salary. A card account in rubles can be used for ATMs and a dollar or euro account can be used to transfer money home.
The type of documents required to open a bank account depends on the kind of services you want, the type of account and the specific bank requirements. For example, Citibank and Raiffeisen will ask for a letter from your employer. At Delta you can open an account automatically with your passport and link it to your home bank.
While some services are offered online, in many cases (at the Moscow International Banks) you will need to go to the bank to open a bank account.
You don´t necessarily need to open a bank account to withdraw money. You can use your own ATM card from home to withdraw money from the ATMs. But be aware that fees for these withdrawals can be rather high.