From GoToCayman.com - the Wiki for Cayman
Taxis are available at the airport and the fares from there are regulated - check with the dispatcher at the kerbside (should cost about CI$20 to most hotels/condos). Note that hotel vans are not allowed to collect visitors from the airport. There are taxi ranks outside the main hotels and at the cruise ship dock.
There are eight minibus routes in operation each with a distinctive colour (coloured circles on the front and back with the route number) and blue license plates. The routes are: 1 (yellow) & 2 (lime green) go to West Bay with service about every 15 minutes and operates from 6am to 11pm. Route 3 (blue) goes to Bodden Town, (service every 30 minutes from 6am to 11pm). Route 4 (purple) goes to East End, 5 (red) goes to East End and North Side, 6 (dark green) goes from North Side to West Bay (service every hour from 6am to 9pm. Route 7 (dark green with white numbers) operates in the George Town area and 8 (orange) runs to Hutland in North Side. Maximum fare is about CI$2.
All routes run to/from/via the depot in George Town situated beside the library. There are a few designated bus stops, but just stand my the side of the road and wave one of the buses down as they approach.