There are lots of ways to get around Madeira: there are buses, you can hire a car, bicycle (if you're fit) or e-bike, hail a taxi, pre-book a tour or transfer - or just walk! Which is best will depend on how many are traveling and where you're going. On the destination pages you'll find some recommendations for specifc journeys, here we'll explain the transport options in general.
There are, inconveniently, 4½ different bus companies that operate scheduled services - the ½ arising because one of them operates both urban transport within Funchal and long-distance services using different buses and a different fare structure... The different companies cover different geographical areas, but their routes overlap in some places.
Horarios do Funchal (HF) operate the yellow buses that provide local services around Funchal, including the more frequent Linha Verde services that run from the centre out along the Estrada Monumental. For these buses there is a flat tariff, which you can pay on board, or you can buy a multi-journey Giro ticket at one of the ticket offices around town with a saving if you prepay for 10 or more trips. They also operate white-and-yellow interurban buses on routes principally to the north and east of the island.
EACL run mostly between Funchal and the area round Caniço.
SAM run the Aerobus connecting the airport to Funchal and the main hotel district as well as routes principally in the south and east. The Aerobus departs from a dedicated stop outside airport arrivals; in the opposite direction, the stops served are marked with a green sign. There's a second airport bus stop (around 50m to the right of the Aerobus if you're facing away from the terminal) that serves local buses that travel west into Funchal and east to Machico and beyond. Whereas the Aerobus has substantial space for luggage, the local buses generally do not.
Rodoeste serve the north and west of the island.
Bus stops will be marked with the name of the operator whose buses stop there - some will serve several. At the time of writing, both HF and EACL route information is available to Google, meaning that bus stops appear on Google maps and their timetabled services will show up if you ask for directions by public transport. HF, in addition, provide a waiting time service - each bus stop has a number (beginning with H) and if you enter the number (omitting the H and ensuring any letter is in capitals, such as 51D), it will give you the estimated time of the next buses. Rome2Rio offers a journey planner that covers all bus services. It is worth cross-checking any third party information with the operator's own timetables to ensure it is current. There may be promotional or day tickets available - check the operator's web site for details.
Be aware that some bus journeys can be very long as they take the old roads (where people live) rather than the new network of tunnels and may be very infrequent: try not to miss your bus, particularly if its the last of the day! Whereas some walking routes are conveniently accessible by public transport, many aren't: you'll have to choose whether your destination or mode of transport is most important.
HF provide a useful guide to all bus services on the island for visitors.
Taxis and Ride Sharing
You'll find officially-licensed yellow taxis at ranks in many locations which operate metered fares. On a slow day, taxis may come up to groups of tourists at bus stops to popular destinations and offer fixed-priced rides. You can pre-book a taxi either by telephone or through the Taxi in Madeira app which will often be cheaper than a metered journey. Bolt operate in Madeira and at present have integrated their app with the licensed taxi operators so you can choose the service you want from available vehicles.
Like many destinations, Covid travel restrictions have led to a shortage of hire cars and consequently to steep price increases. There's not a shortage of hire firms on the island - but you'll need to shop around. Also note that the online "reservation" system offered by some smaller hire companies may simply be a request form that gets e-mailed to the office. Use websites to compare prices, but you may need to ring up if you want instant confirmation.
Driving on Madeira can be a challenge - the topography doesn't allow for much space for acceleration lanes on modern roads and the old roads can be extremely steep, winding and have deep gullies - or even sheer drops - at the sides. Navigation software varies widely in its understanding of the suitability of roads for vehicles.
Tours and Transfers
It can be more convenient to visit certain destinations by pre-booked tour or transfer. You can get to most popular locations with a shared bus tour (and guide), by private guided tour or by shared or private transfer (you're dropped off and left to your own devices). Many agents on the island will book these services for you.
You can hire road bikes, mountain bikes, e-bikes and associated equipment from many locations. Madeira is not ideal terrain for casual cycling, though there are some dedicated cycle tracks, but it offers all sorts of opportunities for cycle sport.