Cost of living
Like other big European cities, Moscow has lots of places to shop, whether it be the decadent offerings of the GUM shopping centre on Red Square, the local supermarket next to your metro station or an outdoor market (rinok) you can buy food in a number of places and at various prices. Outdoor markets tend to be cheaper than supermarkets if you’re confident with your Russian although gesticulating and intonation can get you far if you aren’t! Here’s a short list of things based on supermarket prices.
You can find information about cost of living in Moscow here: https://www.numbeo.com
Compare prices between Moscow and London here: https://www.numbeo.com
One more website about cost of living in Moscow: https://www.expatistan.com
As for bank accounts, BKC will open an account in your name which you will receive a debit card for. If you’re looking to save money and maybe send it home then there are plenty of facilities in Moscow to do this, whether it be the bank account opened for you, Western Union or Unistream it’s easy to send money home and if you shop around the transfer fees are minimal (e.g. Unistream’s fees are 1.5% of the sum, decreasing as the amount increases).
Visa, MasterCard are accepted almost in any ATM, Visa Electron and Cirrus/Maestro - more rarely, and AmEx and Diners Club owners might have problems cashing the cards.
There're plenty of ATMs (bankomat in Russian) in Moscow, especially in the city centre. However, it's better to withdraw your money from a cash machine located in a bank's office. If your card gets swallowed, the problem will be fixed considerably faster there.
You can easily make your purchases in big department stores and supermarkets in Moscow, as well as pay your bills in a good deal of restaurants.
Some currency exchange outlets will allow you to take cash advances from your credit cards, but it will cost you extra.
Getting your money home
You can officially take up to $3,000 out of Russia with no documents. Amounts from $3,000 to $10,000 will require some additional papers. It's prohibited to take larger sums in cash out of the country.
You can deposit any amount of money to your bank account in Moscow and withdraw it in your home country.
In addition, you can buy travellers' cheques in Moscow and cash them in your home country.