Been there for three holidays in the last 4(?) years, always off-season in Jan/Feb/Mar, apart from last year when Mrs Poacher snaffled a 3-week stay over Christmas and New Year - New Year's eve firework display is legendary, and didn't disappoint.
One of the main attractions is walking along the system of levadas, essentially irrigation canals to transfer water from the wetter North to the drier South (you've probably found this out already from your Portugal book). Many of these are benign and easy walking; some are vertiginous, clinging to the side of mountains to follow the contours, and some are genuinely lethal, with sheer drops of hundreds of feet from a broken concrete parapet with no safety rail. Look on YouTube for e.g. levada do curral, levada do norte if you want to scare yourself. Even some of the regular tourist levadas can have frightening sections - the otherwise benign levada dos tornos has a stretch with ~70 ft drops onto rocks from an 18" wide concrete path if you start from where it leaves a tunnel near Monte (avoidable by a safe but strenuous descent and ascent). If you fancy levada walking (and why wouldn't you?) there are many organised trips, which are great for some of the less accessible walks, but if you prefer to do your own thing I recommend the Cicerone guide by Paddy Dillon and the Rother walking guide by Rolf Goetz. The Madeira Tour & Trail map (1:40,000 scale) by Shirley & Mike Whitehead is invaluable even if you don't intend striking out on your own, just to understand the geography.
Madeira is spectacularly mountainous for such a relatively small island - don't hire a car for exploration unless and until you've got the feel of the place - steep and winding roads. Fortunately the bus services are cheap and wide-ranging (we went from Funchal to Porto Moniz and back via a different route for 12 euros - not far on the map, but 3 hours+ each way, and be sure not to miss the bus, 'cos there won't be another one!). Funchal is naturally the focus of bus routes - if you're not in easy reach of Funchal your options will be very limited. If you're near or in Funchal, get a prepaid
Giro ticket and add to it as necessary; it's much cheaper and easier than pay-per-journey. Bus journeys outside Funchal are provided by SAM to the East and Rodoeste to the West, with some by the Horarios do Funchal which does the urban network. To make sense of the various options,
TJWalking produce a useful summary booklet.
Eating out can be expensive in the evening but reasonably cheap around mid-day - I particularly recommend
Armazem do Sal. Since you're in a villa I'll assume you'll be cooking at home at least part of the time - locate your local Pingo Doce supermarket for best value.
For road cycling info see
YACF thread. If you want any more detailed info on walk recommendations or birdwatching feel free to PM me.
Various possibly useful links:
http://www.bluedome.co.uk/madeira/index.html
http://www.madeira-web.com/PagesUK/index.html
http://madeiraislanddirect.com/blog/
HTH