I haven't played Civ 4 for a bit, but generally the more cities you have the better. Also, it is a common tactic to build a city with a purpose - either for gold, production, or scientific research. Early in the game you don't want to build your cities too far from your capital as it costs more gold per turn to maintain them the further they are from your captial. Also, it is easier to defend your cities when they are nearer each other from those pesky barbarians!
If the race you choose doesn't start with a Scout, then it is a good idea to build one pretty quick so you can find the best place to build your next city when the time comes. You will also find small villages (I forget what they are called, but they are a gold colour) that will give you gold, sometimes map knowledge, or other valuable bonuses early on in the game.
You can make more money from trade routes between your cities. For this to happen they must be connected via road or river (and Sea later on but requires a particular Technology) - for river connection, both cities must be on a tile next to the same river. When you start the game, if there is a river nearby then it is a good idea to found your capital next to the river for this reason (not to mention that the yield of tiles is usually excellent in these areas anyway).
I vaguely remember always rushing to get a religion on the Technology tree as my first task (I forget which religions, but there are two they you get very early in the Technology tree) - the first player to gain the associated technology will get that religion. Religion helps you increase your culture points which periodically increase the borders of a city (every time your borders expand, you need more and more culture points). I believe religion also gives you more happiness (happy faces in the UI) whcih will be more or less important depending on your race, your civics, and the level of difficulty with which you play the game. But as your cities grow (have more population points - the white circles) happiness will become more of an issue as each population point gives you a negative happy face.
If you found a religion, then it will also spread to other cities. This is instantaneous (I think) for a city connected to your capital via road or river. Otherwise the religion will eventually spread (there is a unit that you can build that will spread your religion to your other cities too). I believe you get a reasonable amount of gold income from having more cities with your religion (you can even spread it to foreign cities if you can enter there borders).
You cannot share food or production between cities.
Scientists (specialists I think they are called generally) allow you more control over how your population points (white circles) are utilised. When you are able to use a specialist in a city then you will be taking away one of your white circles. In other words, White Circles + Specialists = Population points.
I'm a bit rusty on Research, but I'm pretty sure that a portion of your gold from a city determines the contribution that city makes to research? Also, as you say, specialists can also be used to increase research. If you are unsure how changing a population point in a city from a white circle (working a particular tile) to a specialist, then just look at the difference in that cities research contribution before and after you change the white circle to a specialist.
I'm a bit rusty with all this, but it's all coming back to me as I type

Hopefully others can correct any errors i've made.