We highly recommend renting a car for your holiday, as this is the most fun way to explore the island. The roads on the island are sometimes steep, which can make cycling a challenge.
We have had very good experiences with Enterprise Rent a Car. Your car is waiting for you directly at the airport in Funchal and booking via the internet and directly on the spot works without any problems.
Furthermore, Rent a Drive offers a really good service, like free airport pickup.
Madeira Island Motorbike Rental would be a good address if you want to travel with a motorbike.
And Rentalcars.com is the search engine for other providers.
The tour operator "Madeira Acessivel by Wheelchair" offers special tours and transport for people with reduced mobility or for people in wheelchairs on Madeira.
In Portugal you pay in euros. Card payments are possible almost everywhere without any problems. Nevertheless, you should always have some small change with you, for example to feed the parking machines. Attention: Parking without paying can be expensive!