I play with raging barbarians on (makes colonizing the new world a pain in the arse) so my first build is a barracks. After the barracks is done, Archery is done, and possibly mining. So two archers are the next build. At this point my Warrior should be on his way back home, if he wasn't eaten by an animal. The next build will depend on how badly I need a second city. If I have a spot open and need to grab it, I'll build a settler, if not, a worker. Bronze working will be done, or awefully close to it, by the time the worker is done. When Bronze working is done, I look for Copper, if there is no copper easily accessible, I'll begin Iron working, if there is Copper near by, I will go for a situational tech (Agriculture, Animal Husbandry, Sailing) before going to Iron Working. Second city is down, build a barracks there and a new Archer in capital. After the Archer a Settler (if I have a worker already or really need to get that third city), or a worker. On this type of map I've never been able to start with more than three cities, my slow start in city spreading does work against me in that manner. After the cities are down make sure each city has at least two archers in it, four if its a "Front Line" city. By this time, your worker has connected you to your Iron or Copper and you should start turning out either Swordsman (prefered) or Axeman. I cannot stress enough the importance of having that barracks in all your early cities ESPECIALLY if you are not an Aggressive Civ. (if you are aggressive you can have Combat level 2, Shock, or Cover, right from the start) Set your science rate to 80% while maintaining a positive cash flow. Once you no longer have a positive cash flow, cease production and find someone to pick on. Rinse, lather, repeat as often as desired. You should be able to maintain a healthy cash flow without religions well into the Industrial era. Never pick on more than one at a time, if someone joins in, sue for peace with the other one, even if the terms aren't favorable. (just don't lose a city over it)
I generally have four or five war times. First one is with Swordsman. Second one is with Samurai (Maceman) and Catapaults. Third one is with Cavalry and Samurai/Riflemen). Fourth is with Tanks and Bombers. Fifth is with Modern Armor, Stealth Bombers and ICBMs. Most of the time the fourth and fifth merge into one big blur. Mounted units, except for Cavalry, to me are good for only reconning, pillaging and opportunity attacks, I don't ever build a task force around them, although I will put a couple spearman/pikemen in my cities if I know that the other uses Mounted units, especially as their UU.