Walking is a popular activity on Madeira and some of the walking routes can get very busy at peak tourist times. There is a network of official routes that are monitored and maintained and you should check if the walk you plan to take is presently open [you can change the language at the top of the web page]. Note that the information on this page is not always updated quickly in the even of sudden closures - these are often announced via the home page of the IFCN (Institute for Forests and the Conservation of Nature).
Although Madeira generally has a mild climate, there can be sudden heavy rains that can make tracks treacherous - or wash them away completely - or cause rock falls. This can lead to walks closing at short notice. In dry weather there is a risk of fires.
Before setting out, make sure you have protection against both rain and sun - the weather can change very quickly and some areas can be very exposed - as well as adequate footwear, water and food and a mobile phone and - if your levada walk includes tunnels - a torch and head protection. Be aware that the temperature can drop significantly at higher elevations.
Watch your footing - vegetation can conceal the edge of steep drops and pipes and cables often snake loose across levadas - and know your limits.
Walks generally follow levadas (irrigation channels) or veredas (off-road paths). They vary in difficulty considerably so do your research before you set off and don't be afraid to turn back if you're uncomfortable or the weather changes. There are plenty of guided options and a self-guided app if you're more adventurous but need help with the route.
Mapping, particularly electronic mapping, can be very hit-and-miss is you plan on straying beyond marked trails. Google maps shows little more than the entry and exit points of some walking routes and even some vehicular routes don't show up. OpenStreetMap generally has more comprehensive coverage, but the quality and accuracy of the information depends on the submitter.
Wherever you walk, you're likely to find spectacular scenery. This site concentrates on places that are reasonably accessible by public transport. Some of the most well-known and popular trails are not served by public transport at all as there are no population centres nearby, so look on the walks described here as supplementary options rather than an exhaustive list.