On Vila, Luganville, eastern Santo and eastern Malekula there are buses and taxis. But routes vary, often at the request of the passengers and the roads, certainly out of town, are more suited to 4WDs. In fact many taxis are 4WDs. You can hire cars and scooters in the larger towns.
There are passenger boats which go between various islands. Many don’t return to your departure point until the next day or days so accommodation on your new island is necessary. If you have the budget you can hire a yacht which includes skipper and crew.
Air Vanuatu is the only domestic carrier with planes seating 9, 20 and 44 passengers with services to some 30 islands.