The concept of babysitting is a bit different in Russia than you might be used to in your home country. Typically, babysitting is done by a family member or a close friend, who is not paid for these services. It is unusual to expect to pay someone you know for babysitting services. For example, in the United States it would be common to ask a friend’s teenage child to babysit – in Russia, this would create an awkward situation because the teenager may feel obligated, but would not feel comfortable accepting money for the services.
For regular babysitting services, one good way is to hire a babysitter (who will be referred to as a nanny) through an agency. However, you will need to establish a fairly regular schedule – for example, twice per week, Fridays and Saturdays for four hours – to be able to secure services, and make a long-term agreement to make the agency commission, which will be several thousand rubles, worthwhile. See Infants > Nannies for names, rates, and contact information of several agencies.
Another good alternative is to liaise with fellow expatriates who can provide a referral. They may have a full-time nanny interested in making a little extra cash in the evenings, or a part-time nanny in need of extra hours. Some young college students familiar with the western babysitting practice may also advertise their services on forums targeted at expatriates. http://www.expat.ru/forum/family-children/ is a good place to look for referrals, possible candidates, or to post a request for a referral.
The standard rate for babysitting is 100 – 150 RUR per hour for a nanny hired through an agency (plus the agency commission). Rates are generally about the same through private arrangements, without a commission.