There are many opportunities to rent a car in Croatia - at the airport iyou'll find several rental agency right when you come out of the arrival hall. The prices are almost the same, so if you have a rental agency that you prefer you should rent your car with them.
You can also book your car at home - we have really good experience with Economy Car Rentals. They cooperate with a lot of different rental agencies - it's easy fast and cheap compared to some of the major suppliers in the market.
You should be aware that if you rent a car and there is an injury to the car, then you should contact the police, whether you are at fault or not. You should also contact the police if you for example get an injury in a car park.