Monarch is my standard difficulty level now, so all of this should relate well.
First of all, while a lot of the advice you've been given above will
sometimes -- maybe even
usually -- work, there is no such word in Civ4 as "always." It all depends, and you need to consider:
1) Who are you playing?
2) Who are your neighbors?
3) What kind of space do you have, and what kinds of resources?
4) As the game progresses, how does the diplomatic situation evolve?
I don't have a set strategy; I have algorithms, if-then statements, and each of those questions is important.
For example, my early-game strategy
usually goes: explore, build a city or two, build the Oracle to get COL to found Confucianism; BUT -- there are at least two exceptions. Does my civ start with Mysticism? If so, then I'll try for Hinduism immediately, and if I get it I'll tailor the Oracle towards getting Metal Casting instead. Do I have some AI right close by, with copper readily available? (Or horses if I'm playing Egypt or Persia, or neither if I'm Inca and my too-close neighbor isn't Mansa Musa.) And especially, did said neighbor found a religion himself? If so, I'll forego the Oracle altogether and do an axe rush instead.
(Note: one of the more nonsensical "always" ideas you'll find is "always find an AI and do an axe rush." Axe rushes are very situational. Why bother if your closest neighbor is comfortably far away? But on rare occasions, it is called for.)
When do I start my warring, assuming no axe rush? Well, that depends. It's very situational. Usually when I play a leader with warmongering traits (aggressive, protective, or charismatic), I'll do my first one once I have Iron Working and Construction, BUT -- my current game I'm playing Lincoln (Phi/Cha) and because of the position I had on my continent I found myself expanding peacefully and had all I could deal with economically just from founded cities and barb towns, so my first real war used Rifles and Cannon -- AND (on the other hand), a previous game I played as Boudica saw its first war before I had any catapults, but that's because everyone else on the continent declared on me.
It's just not predictable, and you have to have various paths depending on what gets handed to you.
Found a religion? I say yes. Unless you end up doing an axe rush and take someone else's holy city, which won't happen most games. A holy city in the early game just makes things so much easier. Also, if you found a religion you're guaranteed to have one, and don't have to wait for the unpredictable arrival of someone else's faith.
Building wonders? For me, almost always the Oracle like I said. What else? Well, it depends. Am I Industrious? Do I have marble? Is my neighbor a nasty warmonger so that I need to put my effort into the military instead? What kind of victory am I shooting for? Each victory requires a slightly different approach.
Tech trading has lots of potential that many players don't realize, and this greatly boosts the value of the Oracle or being the first to Liberalism. Example: In my current game, I was behind a lot of techs, but won the Lib race and took Astronomy. I then had three expensive techs, Education, Liberalism, and Astronomy, that other civs would trade for. Trading these repeatedly got me Banking, Gunpowder, Guilds, Engineering, Economics, Nationalism, Printing Press, and Constitution. Researching those two techs, in other words, got me eleven techs. Not too shabby, eh?
Also, don't neglect the possibilities of trading techs for cash. If some AI is really backwards but has a lot of money, so that you don't mind tossing them a technological bone, call him up, put a tech in the trade window and ask "what will you give me for this?" Typically, the AI will offer his entire bank account, maybe with a world map thrown in just to sweeten the deal. This can come in handy when you're doing massive upgrades.
When the AI demands tribute, do you pay up or not? Remember that when you do, you get a 10-turn peace treaty and neither of you can declare war. You also improve that civ's attitude towards you. Worth doing? Very situational, depending on whether they can attack you, whether you want to go to war with them, and what they're asking for.
Anyway, I could go on forever, but I hope the point is made: there are no hard and fast absolute rules.