How exactly does caravan trading work?

I'm kind of inexperienced on how the caravans work. There are two arrows when you select a caravan: one that points to the left screen and one that points to the right.

As far as I know, the left point is what you get from the trade and the right is what the city gets.

Am I correct about this? Could somebody explain this to me?

Correct, what you get is the left side and what your partner gets is on the right

What about sending food, production, etc.?

Will food be taken from the home city and given to the receiving city?

You can send food when you have a granary in the starting city and you can send production when you have a workshop in the starting city too.
Food doesn't get taken away from the starting city, the receiving city receives food.

Also worth noting that Cargo Ships give much better returns than Caravans. If possible you should always use them by preference.

You can massively speed up the early growth of a city with a food cargo ship.

As a rule of thumb I try, if at all possible, to get a coastal capital and one or two other coastal cities, then to spend the early game with food going back and forth. The extra population speeds everything else up.

Resource caravans I find less worthwhile, though I'll use them if my growth is outstripping my happiness.

Overseas caravans I tend to find worthwhile only in the late game, when money is more important than growth, or if there's some sort of extra bonus in it, like city state influence or critical religious pressure or the tourism boost. This isn't universal though, and at the highest difficulties the extra science gained from caravans to other civs can be critical. At anything from Emperor downwards, however, the science yields tend to be minimal for most of the game because of the AI not having a tech lead over you, making food a much better proposition.

Does sending food always work? I tried sending 2 caravans to my capital that sent a total of 8 food and it still was in stagnation.

oPunchDrunko said:

Does sending food always work? I tried sending 2 caravans to my capital that sent a total of 8 food and it still was in stagnation.

Click to expand...

Yes. It works. Check what tiles your capital is working. If you don't manually assign, check what bias you have it set to.