Welcome to civfanatics, whkoh!
What determines the speed at which workers build improvements?
First of all, the game speed setting. Improvements take 50% more time on epic and 200% more time on marathon. They take 33% less time on quick setting.
Next, the type of improvement. Some improvements are quicker to construct than others.
basic time needed:
road: 2
railroad: 3
farm: 5
mine: 4
workshop: 6
lumbermill: 8
windmill: 5
watermill: 8
plantation: 4
quarry: 6
pasture: 4
camp: 4
well: 10
winery: 5
cottage: 4
fort: 10
remove jungle: 4
remove forest: 3
scrub fallout: 6
forest preserve: 8
When you build an improvement on a tile which contains a forest or jungle and the improvement cannot coexist with the forest or jungle, then the time to remove the forest or jungle is added to the time to build the improvement.
The speed of your workers is increased by 50% when you use the serfdom civic. The speed of your workers is also improved by 50% when you invent steam power.
So a worker building a workshop (base time 6) at normal game speed but with the help of steam power, would build the improvement in 4 turns (6/1.5) as it is constructing it at 150% speed (100% + 50% bonus from steam power). Build times are rounded up.
The time to build an improvement should not be an issue. You should always build large numbers of workers so that your citizens can work on improved terrain. Improved terrain is much more efficient than unimproved terrain.
Why can't I build a farm? No reasons given in the dialog; the option's just blacked out.
There's no small pop up information telling you that you need civil service to build the farm when you hold your mouse over the 'build farm' option of the worker?
Some explanation about how farms work:
Farms add 0 food to a tile in basis. This is improved to 1 when the farm is irrigated. A farm is considered irrigated when:
1) It is build on a flatland tile (flood plain/plains/grassland/tundra) next to a fresh water source (inland lake/river/oasis). You can recognise the fresh water tiles by holding your mouse over them. It should tell you 'fresh water'.
2) It is build on a non-tundra flatland tile next to (diagonally, horizontally or vertically) an already irrigated tile after the discovery of civil service. Civil service is the technology that allows the spreading of farms. A centre city tile can take the place of an irrigated tile in such a chain of irrigated tiles, although it has to follow the same rules. So a centre city tile on a hill doesn't spread irrigation and a centre city tile on a tundra tile doesn't spread irrigation when that tundra tile isn't directly adjacent to a fresh water source.
When you build such a farm and you hold your mouse over it, then it tells you 'irrigated'. It also looks visually different to a non-irrigated farm.
This food output is further increased by 1 after the discovery of biology. Some food resources further increase the food output as detailed in the civilopedia (1 extra for wheat, corn or rice without farms, 1 further extra for rice when farmed, 2 further extra for wheat and corn when farmed).
Farmed and irrigated grassland wheat tile post-biology: 2 (base for grassland) + 1 (wheat) + 2 (bonus for farm on wheat) + 1 (irrigated) + 1 (biology) = 7. This is the highest food output in the game (under normal settings, some weird settings allow wheat on flood plains).
You cannot build a farm when it would add 0 food. So that's a flatland square not connected to another irrigated tile or fresh water source. Tundra tiles non-adjacent to a fresh water source or hill tiles or ice tiles also cannot have a farm.