Madeira is awesome for birdwatching!
SPEA the Portuguese Bird Society have published a booklet amongst many other publications on birdwatching all over Portugal .It has a section on Madeira with much info including location coordinates
Ponta de Sao Lourenco
Foz da ribeira da Machico
Lugar da Baixo..south west coast
Ponta do Pargo and Porto Moniz…south west and north coast
The Laurissilva Forest
Fajã da Nogueira, Balcões, Folhadal,
Rabaçal, Fanal, Chão da Ribeira and various
viewing points all around the island.