Madeira - Funchal or elsewhere?

We are hoping to go to Madeira next year.
I'd welcome your thoughts about where to base ourselves.

We will be going for at least a week, maybe longer.
All adults, no children.

We enjoy walking and sightseeing.
We would want to be able to walk to a variety of restaurants in the evenings, generally prefer somewhere different every night.
Also like nice bars before/after dinner but don't need nightclubs.

Funchal seems like the obvious choice but would welcome other suggestions for other places to consider.

We may hire a car but depends where we end up staying and how easy it is to travel around from there.

Also any recommendations for fabulous hotels (in Funchal or elsewhere) would be very welcome. We would like somewhere with lovely outside space to sit, ideally a pool but as we may go in Spring it might not be warm enough so not essential if it has gardens.
We like a decent sized bedroom with some personal outside space (balcony or patio).
I know sometimes you need to be outside the main towns to fulfil that brief, we're happy to keep an open mind.
Would consider self catering (I prefer it in someways so we can sort our own light breakfasts, and lunches on days we stay local, and eat out in the evenings.)

Help with any or all of the above much appreciated.
Thanks