Hi there and welcome to Civ and CivFanatics! This community is awesome, I hope you enjoy your stay.
In the games you're used to, resources are usually treated as a sort of cash - you gather some amount of it (from a gold mine, Vespene geyser, w/e), and then spend it to create units and buildings and to research technologies. Civ, however, has several different types of resources, all handled differently.
First, there are the "tile yields". These are found on almost every tile in the game, and are represented by

(food),

(production), and

(commerce) icons. You can see these icons in the city screen (double-click one of your cities), or by pressing Ctrl+T (I think...). You "acquire" these yields on a per-turn basis. Every turn that a citizen (white circle in the city screen) is working a tile,
that city (not your empire as a whole) gets the benefits of that tile.

is required to grow your city (for more citizens, allowing you to work more tiles);

is used to create units and buildings; and

I'll explain in a minute.
Second, there are the "resources". These are fewer and further between, but you should always try to found your cities near at least two or three of them. You can press Ctrl+R to see little bubbles pointing them out. They include food resources (Corn, Deer, Pig, etc), happiness resources (Spices, Gold, Dye, etc), and strategic resources (Copper, Uranium, Marble, etc). Resources have two purposes. First, their tile yields (once your Worker builds the correct type of improvement on them) are
much higher than those for normal tiles. (To give you an idea, a riverside grassland farm yields 3

1

; a riverside grassland farm on Corn yields 6

1

!) Second, each resource carries some kind of global bonus to your cities. For instance, once you have built a Mine on some Copper (and built roads connecting the Copper and all of your cities to each other), you will be able to build Axemen and Spearmen (vastly superior to the Warriors you can build at first) in
all of your cities - not just the one closest to the resource. Similarly, happiness resources like Silver or Incense will give +1

in all of your cities; and health resources like Rice will give +1

in all of your cities.
Finally, open up the city screen and look at the top left corner. You should see some of the following icons:

,

,

, and/or

- next to + and - buttons. These are your
sliders. All the commerce your cities generate is fed through these sliders to determine what you do with that commerce.

is used to research new technologies;

(culture) expands your borders (and the culture slider also makes your people happier);

is some arcane feature I know nothing about because I don't have the expansion packs

; and

(wealth) turns your commerce into cold, hard cash for you to spend. However,

isn't as obviously useful as you may think. For most of the game, it's basically used as trading material with other civs, and to have a buffer so that you can push up your research until you're running at a deficit. If you build the Pyramids or research Democracy, though, you gain access to a civic called Universal Suffrage that lets you use

to rush the production of units or buildings, completing them several turns ahead of schedule.
I hope this helps - one of the toughest things for an RTS player to learn in Civ is how the frigging resources work. Believe me, I know.
EDIT: Go here:
http://www.civfanatics.com/civ4/strategy/introductory_courses