I played a game of Civ IV vanilla before I learned that the full version comes with expansions I hadn't been using. I also noticed something very quickly: despite connecting my cities to one another and hooking resources up to them as well, doing so brought me no benefit to productivity/food despite the game implying that this should be the case. It's actually hitting me quite hard, because I have a coastal city that'd be a great and booming place if not for its production value being a lousy 1; I have the copper and iron and marble to technically raise it, but somehow it just won't work.

Am I doing something wrong here? What is going on? I can provide pictures if needed.

What terrain tile are the population working?

I have them work whatever special resources I build the appropriate improvements for first, since those tend to be way the fuck better than alternative tiles might be. I can even see that my cities have access to the special resources in the screen to the right, but both the production and food totals I have genuinely aren't adding up well.

Connecting a resource to your network will bring a special benefit (happiness, health, ability to produce certain military unuts, doble production for some wonders/buildings) to all your connected cities. That's what the columns on the right in the city screen mean. In order to get the tile yields (food, productions or commerce), you need to work the tile (the center of the city screen) and it only applies to the city that is working it.

(Later in the game, with the Beyond the Sword expansion, corporations will produce yields if resources are connected, in exchange for money).

Yes, I understand this, and this isn't my issue. My issue is that in vanilla Civ IV and according to the civilopedia, getting resources also gives the benefit of +1 food or +1 production to each city that has it, but I haven't been noticing such benefits at all and I don't know if the issue is me or the game.

I haven't played vanilla in a long time, but I have never heard of a global food or production bonus. I would have to check the Civilopedia though.

It gives a bonus to the maximum capacity in health or happiness. In effect these specials allow each of your cities to to get bigger.

I launched Vanilla and took a look at the Civilopedia. As you can see, pigs give:

+1 food to the tile they're in, without improving it (only applies to the tile and is only collected if worked). For example, in my starting location, there are pigs next to Athens in a grassland hill. Because there are pigs on it, this tile yields +2 food and +1 production, instead of the usual +1 food +1 production for grassland hills.
+3 food if a pasture is built (needs to be worked as well).
+1 health if a pasture is built (in this case, to all cities that are connected to the resource network).

You are misreading this; a resource like iron gives +1 production only to the tile it's on, not to every city. Later in the game corporations can work like this (the more iron you have, the more productivity bonus mining inc gives) but for the vast majority of the game the productivity bonus of resources is restricted to the tile they're on.

With that said, strategic happiness resources such as gold can be thought of as productivity bonuses. This is because they help control unhappiness from slavery, and slavery is the most efficient way to quickly build things. In the early-game you want to settle nearby food resources so the city can grow quickly, then use slavery to convert that growth into production.