The Madeira Botanical Garden was created in 1960 and has become a major visitor attraction, often coupled with a cable car trip.
The garden is divided into a number of areas including native plants of Madeira, succulents, palms and cycads, medicinal and aromatic plants and commercial species. There's also a small natural history museum.
Apart from the gardens themselves, there are great views over Funchal and the surrounding area. Entrance is quite reasonably priced - around €7.50 at the time of writing.
More information on visiting can be found on the garden's own website.
There is car parking opposite the garden.
HF bus 31 runs to the main entrance to the gardens and terminates there. The 31A goes to the main entrance, but also runs around to the cable car station which is a little higher up. The 29 will also take you to the gardens, but runs on out to Curral dos Romeiros, so ask to be put off at the botanical gardens if you are unsure of the stop.
Taxi drivers may pull up at bus-stops if they see a group of waiting tourists and offer a fixed price trip.
You can combine a visit to the gardens with a cable car ride - but note that from Funchal you need to take a cable car up to Monte and then another one back down to the botanical gardens. Or you could take a single cable car ride up from the gardens to Monte and come back down on one of the famous wicker toboggans!
There is a (strenuous) walking route from Monte down to the Levada do Bom Successo which ends close to the botanical garden.