I think there are 3 key early game concepts that need to be understood and implemented fairly well on monarch to win, but can be done relatively poorly, or even not at all on prince and still win with relative ease.
First ... quick expansion to 2 or 3 cities. Settlers are very expensive. Building them stunts your growth. To get that second and third city out quick, you need to get BW so you can chop some forests. 2 forest chops to shave 10-12 turns off that first settler is more huge than you probably realize. It means 10-12 more turns of growth for BOTH cities, plus 10-12 more turns of production in BOTH cities.
Second ... choosing the correct tech path. Unless you really know what you are doing (I can't say I am at that point yet), it is generally unwise to waver from the following teching plan:
Necessary worker techs first, usually only 1 or 2.
Bronze Working
Writing -> Alphabet
BW is your key early game war and production tech. You want copper, you want the whip, you want to chop.
Alphabet is the next key. It allows you to trade and fill in everything you skipped. Even on Monarch, if you get Alphabet quickly according to this plan, you will be first, and probably half the AI won't even have writing yet. Staying ahead at Monarch generally requires researching beelines along techpaths the AI doesn't favor and filling in the useful techs you missed by trade.
Third ... Getting your economy up and running early. Hammers were coin of the realm in earlier civ games. In civIV, it's food and commerce. Yes, you need production cities, but 2 of your first 4 cities should be high-food commerce cities. Cottage economy is easiest. Pick 2 cities with lots of grasslands/floodplains and a nice food surplus, and put building and working cottages at highest priority. And make sure you have enough workers for this.