So, I played as the Russians.
Here are my strategies:.
Russians are the expansionist civ (it was well identified by a poster earlier). Knowing the lay of the land is important.
So your build order is to build 2 scouts, and then whatever else. Grab all the goodie huts you can get and talk to all the city states you can talk to. Take note of all the nice resources around, especially luxury resources. DON"T FORGET! YOU GET +1 PROD on Strategic Resource tiles! Production is SHY in civ5, this is actually huge! SO GO GET 'EM!
So 1st policy to grab is the Liberty. You need to be cranking out the settlers.
1st tech to research, I recommend Animal Husbandry. You need to see where those horses are.
Don't be shy. Be a push-over. If there are some good spots to settle and its close to someone else, go for it.
Try and claim as many resources as you can, as fast as you can.
Capital build order:
1. Scout
2. Scout
3. Worker (if not at liberty) but switch to Settler (as soon as liberty is adopted)
4. Finish up that worker
5. Settler or Granary (depends on resources availability)
6. Settler or Granary (same as above)
7. Military
2nd city build order:
1. Monument or Granary (have not decided which is best) - I think Granary as it allows you to put in higher production tiles
2. Military
3. Worker
4. Monument
3rd city build order:
1. Granary (I have decided now that this is better than Monument)
2. Worker
3. Monument
4th city build order, and the rest:
1. Granary
2. Monument
3. Military
4. Military
5. Whatever
Social Tree:
Liberty tree as a priority, fill out until you get +1 prod in each city. You are going to be big, it helps.
Piety is also very good!
Tech Tree:
Animal Husbandry 1st, as I said. Then try and get the techs that allow you to exploit the luxury resources you have. You're big, so you need them. Top it off with currency for market and the +25% gold.
Military:
You're focusing on expansion so military may be lacking early on (albeit AI sucks, so you'll have nothing to worry about), however once you get the strategic resources you will crank out those resource-based units fast (I had 24 iron with only 2 iron deposits!)
City States:
Don't bother giving them gold! You need gold. If you want to befriend them, do missions only. Gold is yours!
Diplomacy:
Settle early to other civs borders to lock them in. AI will just ask you not to settle anymore. Agree to it, you got the resources anyways.