Dolly's Auto Rental and Sleepin Guesthouse rent cars (from G$7000 per day), but with bad roads full of farm animals and crazy drivers, few people decide to drive themselves in Guyana. If you do decide to drive you will need a Driving Permit (available for free at Cheddi Jagan International Airport arrivals terminal).
For simplicity and safety, taxis are the way to get around central Georgetown; trips around the center cost G$300 to G$500 or so, even at night. Have your hotel call a reliable cab company. If you need to flag down a taxi, use only registered ones painted yellow (all registered taxi license plates start with a 'H') and try to find ones with a company logo on the side. Recommended operators include GR Taxi Service and Cyril's Taxi Service.