heavily, intensively
The land has been intensively farmed.

organically
They farm organically now.

Farm is used with these nouns as the object:
fish, land, salmon, sheep

big, large
little, small
100-acre, 200-hectare, etc.
local, nearby, neighbouring/neighboring, surrounding
isolated, outlying, remote
hill (BrE)
family, home
conventional, traditional
private
collective, cooperative
commercial, corporate (AmE)
state
factory, industrial (esp. AmE), intensive (esp. BrE)
experimental, research (esp. AmE)
model
Ten model farms have been set up to showcase modern production methods.

working
The area combines a working farm and a farming museum.

organic
arable (BrE), livestock, mixed
breeding (esp. AmE), horse (esp. AmE), stud (esp. BrE)
chicken, coffee (esp. AmE), dairy, fish, fruit, hog (AmE), pig, poultry, sheep, shrimp (esp. AmE), tree (esp. AmE), trout, turkey, etc.
truck (AmE)
wind
controversial plans to build a wind farm on the island

have, own
manage, operate, run
work, work at, work on
During the war, few men were left to work the farm.
The children had to work on the family farm.

build, establish, set up, start
live on
be raised on (esp. AmE), be reared on
I was raised on a country farm.
The pigs are raised on factory farms. (esp. AmE)

inherit
visit

be located, lie
The farm lies on the hills above the lake.

grow sth, produce sth
a farm growing tobacco
a dairy farm producing gourmet cheeses

operate
The 80-acre farm now operates around the clock.

produce, product
Farm produce, including fruit and corn, was their principal export.
farm products such as eggs and vegetables

animal
labourer/laborer, worker
labour/labor
chore (esp. AmE), work
I helped him with his farm chores.

manager, owner
management
field (esp. AmE), land (usually farmland)
building
equipment, implements, machinery, tractor
policy (esp. AmE)
bill
Republicans hope to pass the farm bill in the current fiscal year.

program (AmE)
the US farm program for wheat

subsidies
The EU has decided to cut farm subsidies.

exports, production (both esp. AmE)
We anticipate our overall farm production next year to be lower.

income
Farm incomes rose 11% last year.

economy (AmE), sector (esp. AmE)
The farm economy was stable.
Today, the farm sector employs about 3% of our workforce.

life
village and farm life

boy, girl, kid (informal, esp. AmE)
family, household (both esp. AmE)
a poor farm family

cottage (BrE), house (usually farmhouse)
shop (BrE), stand (AmE), store (AmE)
The farmer's wife runs the farm shop.
Bo was working the farm stand.

supply store (AmE)
Local farm supply stores are expected to lose business.

community, country, state, town (all AmE)
the rolling farm country of central Kentucky
an Illinois farm town

belt (AmE)
the Kansas farm belt

road, track (esp. BrE)

at a/the farm
The police are investigating a fire at a nearby farm.

down on the farm
The myth of happy animals down on the farm is now far from the truth.

on a/the farm
He had lived on that farm all his life.