Getting around the Comoros requires patience and flexibility. There is no public transport system to speak of, and schedules are more like suggestions. But the islands are small, and with the right expectations you can reach everything you need to.
Between islands, your options are small planes and ferries. Domestic flights connect Ngazidja, Anjouan, and Moheli on small 10-20 seat aircraft, with flights typically around 30 minutes and costing roughly 50 euros one way. Book well in advance during dry season as planes fill up fast, and be prepared for cancellations and delays -- they are routine. Baggage limits are strict, usually 10-15 kilograms. Ferries (motorized boats, really) run between the islands for 15-25 euros, taking two to four hours depending on the route and conditions. Ferries are less reliable than flights, especially during the wet season when rough seas can cancel sailings for days. Buy tickets at the port and confirm departure times the day before -- they change constantly.
On the islands, shared minibuses (taxi-brousse) are the main form of transport on Ngazidja and Anjouan. They run along the main coastal roads, leave when full rather than on schedule, and cost very little. On Ngazidja, they connect Moroni with the main towns around the island. Moheli has very limited public transport -- you may need to arrange a vehicle through your guesthouse.
Taxis are available in Moroni and the larger towns. Agree on the fare before getting in, as there are no meters. For exploring beyond the main roads, hiring a car with a driver for the day is the most practical option and not expensive by Western standards. Self-drive car rental exists on Ngazidja but roads are rough, signage is minimal, and driving standards are creative. On Moheli and Anjouan, roads inland can be very poor.
Mayotte operates as part of France and has a completely separate and much more developed transport network, including regular buses and car rental agencies.