You can get around Madeira many different ways, depending on how much effort you're looking to put in.

Buses on Madeira are reliable and affordable.  There are three main companies that offer services around the island, and all three make stops in Funchal (the island’s capital and main city).  Current, detailed information about routes, fares and timetables can be obtained at most hotels, and in all bus stations. 

Taxis are available in the major cities and tourist areas throughout Madeira.  They can be recognized by their blue stripes and yellow color (see picture)  It is usually easy to catch them around the busy streets, where they will stop when waved down.  Airports are full of cabs and there are many taxi ranks (stands) around the city and hotels and restaurants will arrange for pickup if you need it.  Fares in the city are set by the government, and based on destinations (a list of which can be found on the back seats).  Fares outside of the city are generally based on the kilometer.

Renting a car while in Madeira is a good option if you want a bit of extra freedom and enjoy shifting gears a lot.  There are many rental agencies available around the island.  Driving is done on the right hand side of the road.  Check out Driving in Madeira for more information.

If you're really up for a workout, you can also join one of the biking tours that will show you parts of the island on bike.  Note that you may have to ride next to lots of cars making sharp turns, so make sure you go with a proper guide and that you're aware of your surrounds at all times.

You can book tours directly for almost 50% of the price you may find in a normal tour operator and just doing it through small companies offering prices for groups of 8 or less: 3 walks along the Levadas and ridges and tours around the Madeira Island  for 55€, 4 walks & tours for 70€  or maybe 2 tours for 38€ . That would provide you with a lot of information and familiarity with Madeira.