Easy question,

Do you have to connect improved tiles to the city by road to gain the bonuses from the improvement?

If I have a farm, but its not connected by road, do I get the extra food bonus or don't I?

certainly someone knows this, please tell!

yes you are right. No road needed to get the extra food/production/commerce from a tile...

However, you do need a road to get the benefits of any special resource there, like iron/pigs/wine.

To clarify what he said in case it didn't make sense, if you build a mine on iron, you get the bonus production from the iron, but you can't build units requiring iron until you build a road connection as well.

I'm pretty sure you don't need to build a road to get the resource on your trade route if it's on a tile with a river that connects it directly to your cities.

Edit: sorry no, I should have said the river has to be directly connected to your capital. Having it connect to any of your other cities, but not your capital doesn't work. If the river is directly connected to the capital, then any resource you improve that is directly connected to that river doesn't have to have a road built on it.

Edit again: no wait, it does work. You do not need to build roads on any connecting river that leads to any city within your trade network.

I'm pretty sure you don't need to build a road to get the resource on your trade route if it's on a tile with a river that connects it directly to your cities.

To clarify that even further, a road needs to connect the resource to your capital. Rivers act as roads for this purpose.

For example

[City 1]...river...[resource]....road....[city 2]...road..[capital]

as long as the resource is connected to the river/road network which includes your capital, all cities in that network can use the resource, and it can be traded

river=road for trade purposes