What is the best time of the year to go to Madeira? To go to the beach. And how is the nightlife on Madeira? Where in Madeira can you do all this?
During the whole year it is a good time to go to Madeira. Only during the first months of the year I have experienced some periods with showers, wind and rain. Assuming you are looking for beaches ... Madeira does not actually have a beach scene what most sunbathing beach lovers are used to.
Natural sandy beaches you will not find here easily. Great deal of the natural beaches is rocky. There are however a few (new) man-made beaches, such as Calheta Beach, and the sea pool complexes in Porto Moniz and in Funchal. BUT ... when you are on Madeira you can always visit its little sister-island Porto Santo ... which has 8km beautiful golden sand beach. The Porto Santo ferry line Lobo Marino sails frequently between the two islands.
For nightlife the best place is the City of Funchal. There are a great deal of activity going on that tourists and people from outside the island does not know of ... because it is locally advertised (thus often in Portuguese) however everybody is welcome to participate.
My advice is ... just ask!
When you are in Funchal ... ask any Madeiran (at the restaurant, hotel, taxi driver, personnel at the shop etc.) where and when is the next nightlife activity. The trick is to ask where they would go themselves when going out. You will be surprised the options that are here.