In terms of where to live, expats are generally divided into two camps - those who live out of town in the compounds and those who live in the centre. We live in the centre, because we like to go out a lot in the evenings etc. and it's also convenient for our son's school. Those who live out generally do for two reasons: 1) they want some outside space (although all the compounds only offer shared gardens) and 2) their children go to the international schools in those areas (the Anglo-American School etc.). Really it's up to you, but I can recommend Evans as a very good estate agent, with English speaking agents and reasonable prices (www.evans.ru).
Shopping is a lot easier now. There are the large Mega Malls, which are on the MKAD (the outer ring road), along with Ikea and Auchan (inexpensive food shopping). There's also the new'ish Europesky Mall by Kievskaya Metro and the older Stockmann's by Smolenskaya Metro.
You'll get plenty of advice from the expat groups I posted last night - they all offer newcomer tours and meetings, and the BWC offers an active Family Network for families with children of all ages. The IWC also has a baby group now, which you can sign up to join at their September meeting.
Sorry I didn't finish my post last night - too tired! Hope all this helps and that you settle down quickly and relatively painlessly! There's a lot to do and see here in Moscow, so you won't ever be bored - and there are lots of indoor playrooms now for younger children, which will keep you sane during the winter! During the summer, you can visit all the large parks and estates around town or just pop into your local parks/gardens wherever you decide to live...
If you let me know whereabouts you're living and how long you expect to live here, I can maybe make some more suggestions to help you, like kindergartens etc.