Hello Jonny250, you will love Corsica for natural sport, in summer the bad thing is there are a lot of turists, so that not so wild but if you go to Scandola you will see a fantastic reserve, in less than 10 meters of deep there are all the Corsican species.
Near Scandola you can go to Porto, Piana, Calvi in a few miles you will think you change many time of country.
For cycling Corsica is not the best place, road are slim but for walking there are a lot of way to go, GR20 is the most celebre trip for walker, but there are other like "mare e mare" and "mare e monti" so you can walk accross all the island.
Corsica is a good place for sport and for familly, my english is too bad too explain all what you can do and have to see.
For spearfishing in Corsica in summer it's like in other island of mediterranean sea, you will have to go deep to take good fish, or to go early the morning or in the last hour of the day, fish are shy, slow to come to you and fast to go away, but they are present.