We spent a week there in July. Freeride Madeira for the day would be a top recommendation, we did a day with them using their bikes, and had an amazing day. I recommend booking this before you go.
Plenty of places to eat in Funchal to meet all budgets.
To see the island, I would suggest renting a car even if its just for some of the time you’re there, just make sure you’re ok with driving twisty and pretty steep mountainous roads. Great views though
Again for walking, whilst you can get taxi’s etc to drop you off, a car makes it easier to get to the starting points.
We loved it here, and hope to return soon, so I’m sure you’ll have a great trip.