edit: well that became a massive wall of text quickly. I hope this is of some use to you. I've given mostly general advice and some specific advise regarding the questions you had at the bottom. Now I want to play some civ IV.
The first thing you want to to is survey the situation. Don't determine your strategy before you start playing, determine it while you are playing. Who are you? What are your traits and special units? What resources are nearby? What other civs are nearby? How nearby are they? If you have little space, this becomes more important. You can't see whether there are strategic resources nearby early on, so remove that doubt quickly by researching bronze working, iron working or animal husbandry. Also, build a scout or two. (here, being lucky with your scouts surviving and finding a scout in an early tribal village matters a lot) Especially if other civs are nearby, you might want to defend yourself or attack them. As for your traits, play your strenghts. Financial? Build more cottages. Imperialist? build more settlers. Charismatic? Build stonehenge.
You want to look what important resources you have. If you see gold, silver or gems on the map, that can greatly increase your commerce and thus science early game and give you an extra hapiness. Ivory and fur can also give you extra hapiness. Stone and marble are very useful for buffing the building of wonders. Stone can be used to build the pyramids and get representation early which is very helpful. Marble can be used to build various wonders like the temple of artemis and the great library to give you more great people. You want to make sure you have iron or at the very least horses or copper. If all that fails, ivory. Really though, just get yourself some iron. If you have no strategic resources, or only horse, make sure you get them by trade, conquest, settling, whatever.
As for goto techs and things to do early game. The first couple of techs depend on your situation but I think you want the first two rows of cheap techs pretty quickly. Prioritize the ones that are relevant to your location. If you aren't near water, you don't need sailing. After that it depends on what you want. Do you want to have a lot of cities, than go for currency for the extra trade routes and for courthouses to make sure you can financially support those cities. If you want to have certain wonders, than go for the techs that have wonders, preferably those that you have resources for. If you need hapiness, go for plantations, if you have a lot of the relevant resources or monarchy otherwise. (or the pyramids and representation, if possible) If you have traits that benefit specialists, like philosophical or cultural (doubles library building speed) than get those libraries quick to boost your science and get some great people early on. An interesting tech is the alphabet. If you can get it early on you can be the only one to tech trade which can be a big advantage.
You should trade techs as much as possible. The only reason not to, is to deny specific wonders or units to the AI when you want to build that wonder or you want to attack them.
Also, you'll want to orient yourself on your religion. Do you want to have an important religion? If yes, get a religion, spread it to religionless neighbours quickly so they'll have your religion (which gives an important diplomatic boost), get a great prophet to found a holy building, and work towards the apostolic palace. It not only gives you the ability to screw over AI's you don't like, but also +2 base production on all temples and monastaries of that religion. If your religion doesn't become big, ditch it for the religion most AI's like and/or that has the apostolic palace. The diplomatic bonusses and apostolic palace bonusses are too important to ignore. Alternatively, having no religion can be a great way to avoid pissing off any specific AI.
Lastly, and I don't have ironclad rules for this, but you need to learn how to pace the building of cities early on. Build too many before you have currency and courthouses, and you'll crash your economy (having a solid amount of commerce can aleviate this problem). Build too slowly and the AI's get their grubby hands on all of that good land you meant to settle. For this purpose, I consider currency and courthouses to be important things that I want to have quickly. Alternatively, rushing a nearby AI can work great. Their capital will be a very good city, and you'll instantly have at least a decent amount of cities and some lebensraum.
As for the specific issues you had. If by wealth you mean commerce and science, that's tricky. Perhaps the trickiest part of the game. Part of it is that you need to make long term investments. Investing in either a super city that produces many great people (with the national wonder that doubles great people production and a bunch of wonders and specialists) or in many cities with cottages and good resources, like plantations and moneymines. If you want to have a lot of gold specifically, let me tell you now that gold is of little interest. It's science you want. Gold is a means to science, little more. As for science, having a high amount of science in absolute numbers is vastly more important than the height of your science slider. If it's at 30% but your great scientists are giving you 150 a turn that is much better than 90% with 75 science.
As for growing cities. You need to settle them with at least some workable food resource, preferably more than one. Rice, pigs, fish, crab, multiple flood plains, whatever. If you can't get that, the city won't grow big. Granaries can help out slow growing cities. You'll also want to beef up your hapiness quickly. There are many ways to do that, I've mentioned most above.
