- Research mysticism first to be able to create Stonehenge and obelisks (this will give you culture to expand your borders)
- Move your scout or warrior around the land in order to discover good spots for a new city or to meet other civs (other civs means: go for defense and expand towards the enemy, fill the gaps at the poles and jungle later)
- If both a warrior and a worker cost 15 hammers, go for the worker in order to develop your city sooner. Choose the warrior if that takes 8 turns (so your city can grow to 2 in the meantime)
- After mysticism, research meditation or polytheism in order to get a religion. If meditation is researched while you are still researching it, change it to polytheism (use the tech-screen by pressing F6). If polytheism is researched also, go for iron working. A holy city gives you benefits.
- After iron working, go for alphabet. Once researched, you can trade your techs with other civs. Trade with distant enemies first, you don't want to give your neighbours the edge. Don't trade your alphabet-tech just yet, you don't want others to be able to trade as well, do you?
- Once you discover currency, sell your technology to the other civs. They're glad to hand over all their money to you for inferior techs. Spend this money on research. The first time is the most lucrative one.
- If you have weak defenses, always accept demands from your closest neighbours. Work on your defenses, especially with 'friends' nearby! Bribing won't stop an attack, when they smell blood, you're theirs.
- If there are horses around somewhere, claim that resource!! Iron and copper are around more often (especially near the poles). Research horse back riding only if you have horses. Don't accept it when you trade if you don't have horses, since it's not useful to you. Never trade it with Cyrus, since his unique unit is mounted!
- If you are at war, attack with mounted units and pillage iron, horses and copper resources first, so they can't build strong reinforcements. Try to sneak in a chariot or mounted archer to move and pillage at the same turn.
- Build barracks in every city for a free promotion.
- Choose promotions in accordance to your attacker's army. Choose +25% cover when your enemy attacks with archers for instance or hills/forest defense if you are near hills or forest
This will get you started (I play on noble, will try prince soon)
Edit: some additions, more examples.