Would suggest getting the Cicerone guidebook/ ebook version. Usually cheapest direct from their website.
The only place that is really practical as a base if you are using buses is Funchal. But even then the options for walks using buses is very limited.
Some of the walks have helluva steep drops, so you really do need a head for heights and some confidence.
While the scenery is stunning and I undoubtedly missed out on most of the best walks by relying only on public transport, I am not sure I would go back. I much preferred North Tenerife for walking, buses regular, plentiful and very cheap. A number of places you could base yourself, from small coastal villages to larger towns (La Laguna) and have access to regular buses, shops, restaurants etc.
Similar scenery, but more variety of walks and I much preferred La Laguna as a place vs Funchal. Madeira was also cloudy and wet in the mountains when I was there even when it was sunny in Funchal and weather was very unpredictable.
Also like the look of La Gomera and La Palma for hiking holidays...
If you do go to Madeira, the very short and easy Levada walk from the bus that stops at the trout farm and the descent to Nuns Valley from the bus that stops at the viewpoint/hotel were my 2 favourite walks that you can easily do by bus from Funchal.
You can also look at the Macs Adventure self guided walking holidays website for ideas for walks and if you read the reviews people often say what their favourite walks were.