Here's a few suggestions. You'll almost certainly need to hire a car.
Eira do Serrado, north-west-ish of Funchal. Brilliant views down into the gorge of el Rio Curral des Freiras, and across to some of the peaks. I could have stayed there all day. Cafe nearby.
Drive to Arieiro (at 1818m!), due north of Funchal. There's a cafe etc there. Take the high level path through the mountains to Pico Ruivo (at 1860m), where there is another mountain hut type place which serves drinks. I'm not sure if you can overnight at either location but it might be worth asking.
The path is well-engineered but pretty hairy in places, going through tunnels and running along sheer cliff faces, but it has a reasonable hand rail most of the way. Stunning scenery. I'm not exaggerating!
You can also drive round to Terxeira. It's a long way round, but gives you easy access to Pico Ruivo without too much walking. From there there are further high level paths although i didn't do them myself.
There are also some stunning waterfalls, which are accessible along the levada's. There are a couple in the area around Rabacal towards the west of the island (eg 25 fontes).
Levadas are irrigation channels mostly taking water from the wet north of the island to the dry south. They are almost level so generally good for walking, though in some places they do cross sheer precipices. Where they go through the dense laurel forest (I think it is) you will find that views out are pretty impossible.
Coast-wise, the Sao Lourenco peininsula to the east is very dramatic. It's accessible by road and then a good path.
There's lots more, I'm sure, so enjoy exploring. Oh, don't discount the possibility of snow on the mountain roads, either. It happens every winter.
so very quiet.
