dramatic
daring
emergency
attempted
successful
air-sea (esp. BrE), mountain, ocean (AmE), sea (esp. BrE), water
hostage
helicopter
financial

attempt
Her own boat capsized after she attempted a rescue.

come to, go to, rush to
No one came to their rescue until the following day.

perform
await

attempt, effort, mission, operation
crew, party, squad (AmE), team, unit
a mountain rescue team

service
personnel (esp. AmE), worker
work
boat, helicopter, ship, vehicle, vessel
dog
swimmer (AmE)
equipment
centre/center
an animal rescue centre/center

bid (BrE), package, plan, scheme (BrE)
a financial rescue package for the company

to sb's/the rescue
Her wails of distress brought him running from the house, like a knight to the rescue.

rescue from
his rescue from a burning building

rescue of
the rescue of a man from a burning building

fire and rescue
New Zealand fire and rescue services carried out several extensive searches for survivors.

search and rescue
The navy are on a search and rescue mission.

Rescue is used with these nouns as the subject:
firefighter, helicopter, hero

Rescue is used with these nouns as the object:
economy, hostage, survivor, victim