Dear Shweta,
You are the buyer of the services. Rather than you, the service provider or lessor is supposed to come up with the draft of the agreement. Therefore, tell the car leasing agency not just to provide the car with the key but the lease agreement as well.
Roughly, the draft agreement should include clauses on payment terms, the general process to clear the invoice, fuel expenses, driver's salary if the car is hired along with a driver, accidental damages, damage due to negligent driving, general maintenance, first-line servicing (daily or weekly servicing to be done by the driver), etc.
Thanks,
Dinesh Divekar