Dear Shweta,
You are the buyer of the services. Rather than you, service provider or lessor is supposed 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 also.
Roughly the draft agreement should include clauses on payment terms, general process to clear the invoice, fuel expenses, driver's salary if the car is hired along with driver, accidental damages, damage due to the negligent driving, general maintenance, first-line servicing (daily or weekly servicing to be done by the driver) etc
Thanks,
Dinesh Divekar