Starting: Russians begin with more of the map visible
Ancient: +1 food from plains
Medieval: Defensive units receive the Loyalty upgrade
Industrial: Half-price Riflemen
Modern: Half-price Spies
You start with a portion of the map, makes it easier to know which way to explore. Try to get as many huts and barb villages as you can, and if there are natural chokepoints put a unit or two there to avoid the AI to take your space.
The +1 food from plains will allow for rapid growth of your cities, which can let you settle a lot of land quick, especially if you beeline for code of laws and go into a republic in the settling phase.
Your loyalty on new units in the medieval times should keep your cities safe, even with mediocre defenses, and by this time you should have a good position to survive past the 1600's and choose any win you would like.
additional tips:
1) Specialize your cities, due to the trait one city should be placed close to additional food resources for rapid expansion.
2) A couple of cities along the coast for science and gold. (science city get's library, university, courthouse and harbour, while library and university is replaced by market and bank for the gold city).
3) One good production city with barracks to make veteran units, and then army this up with two non veterans and you'll still have an army. So there is no need to have barracks in more then every third city at the max. This city will also be your wonder city, but keep in mind that if you get a good gold city, rush buying a wonder like colussus, trade fair or east india will help a lot. (The reason I say rush buying these are that the Trade Fair and colossus needs to be in your number one gold city, as it's a city specific bonus. The East India can be placed anywhere and can if low on gold be built in your production city. Units can be rush bought from the other cities with the money you save)
Also when rush buying try this priority on money:
a) economic buildings first due to they pay them self back the fastest: Market, Bank
b) science buildings to keep up on tech
c) Culture and growth buildings, and especially courthouse is nice if you can get a special resource or two by pop rushing it, or the population is already high. Remember since cities are specialized it's important to have as many good commerce/production/food tiles available as possible for the specialization the city has, as the adds up due to bonuses.
