Kitesurfing in Fuerteventura - Matas Bay is the place to be
Fuerteventura is a place with a lot of stable wind so it's ideal for watersports like windsurfind and kitesurfing. Personally, i would recommend the south of the island as it is the home of a world championship in windsurfing and kitesurfing every year and the wind is usually better there. The main season is in the summer when the wind is the strongest, but the schools are open all year round and offer other activities like surfing, SUP or snorcheling( at least Matas Bay does, or you can find specialized surf schools like Rapa Nui) when there is no wind.
That is what I like about Fuerteventura and Costa Calma in particular - usually, when there is no wind there are good conditions for surfing at the west coast of the island at La Pared which is 10 minutes drive from Costa Calma, so there is always something to do. Kite centers are on the east cost, because there are little to no waves.
There are lot of centers which offer lessons and rental, but the one I liked the most is Matas Bay just outside of Costa Calma. It has great atmosphere, really friendly people and for most of the season flat water and good wind. The prices are good compared to other centers - I took a kitesurfing lesson with Andrzej after taking one in the bigger center in Costa Calma and there was a big difference. Andrzej is really good at explaining stuff and thought me how to water start in an hour. On the other hand, the other center didn't even take me to the water eventhough I wanted to and spend 3 hours flying the kite on the beach. Mine biggest worry about kiting was that i wouldn't be able to get back to the shore. Matas Bay has two jetskis and a boat that can pick you up and enough stuff to notice when you go too far, so there is nothing to worry about. I personally got picked up like 5 times, after that I somehow managed to get back to the shore so I think it's a really good spot to learn.
They also offer individual courses for those which want to learn really fast. The only thing is that it's not too easy to find - you have to get offline the highway before Costa Calma to La Pared, then go under the highway and then take the second exit and cross the road to an unpaved road towards the sea. On the other hand, the place is not crowded and really nice even if you want just chill on the beach or just surf on their free wifi :-).