Right now, though, I'll answer a question about doctors here because it's of larger interest. I don't think Moscow is typical of Russian practices, though, so keep that in mind... Russians I know here tend to take their kids to Children's Clinics, where you might usually see the same doctor, or you might not... When your child is sick, you often can get a house call through such clinics. The prices are VERY low by Western standards and the care is generally good, depending on the clinic.
I went to such a clinic, one of the best in the city, when we first moved here in 2004 and Natalia had an acute ear infection. Natalia was a year and a half and still nursing (I had chosen to start weaning her once she had gotten over the stress of our international move and had a routine established). Nursing was the only thing that seemed to relieve her ear pain, so when she started to shriek in the doctor's office, I discretely nursed her. That's when the doctor literally ripped Natalia's head off my breast and SCREAMED at me for nursing her at that age.
Natalia, and I, were LIVID. While nursing at a year and a half certainly isn't common in the US, it's not unheard of... And our Brooklyn pediatrician said it was endorsed under US guidelines for the first two years... It goes without saying that I never went back to the Russian clinic.
Adults tend to have a doctor they generally see who can also make house calls, or they go to a general clinic. At such clinics lines are generally long... As in all areas of life in Russia, connections and greasing some palms speed up the process.
Because we're foreigners, we have medical insurance that covers Western doctors—so we go to one of the clinics that accept foreign insurance and employ a mixture of foreign and Western-trained Russian doctors. Many Russians also go to our clinic, but it's too expensive for your average Russian.
Another common practice here is to pay for a yearly "membership" at a clinic, where they treat you for pretty much whatever comes up. There are rules, though, and I bet the memberships are a lot like insurance policies—with maximums and excluded procedures.