Kitespot Chamey's Nautica
On the east side, you'll find Chamey's Nautica, one of the most popular spots to kitesurf. It is man made, but a paradise: A huge playground with flat to choppy water, and waves further out. At low tide, the water is standing-depth until almost 100m from shore, so a great place for beginners. The kite launch takes a little getting used to (as the beach is only 25m long with palm trees close to the shore), but the locals will happily advise you; and after a couple of days you'll be laughing. The wind here is usually side-onshore and is stronger further out since the beach sits in a slightly protected corner.
If you are a beginner kite lessons are offered for around $120.000 COP (approx 60 USD). The place also holds beach chairs, lounge chairs, a little shop to buy a sandwich and a soda and offers as well rental for kayaks, stand-up paddle boards and many more. A compressor for inflating the kite is also provided, so no pumping required, but be aware if you spend most of your day there (which you most likely will) the owner will charge $10.000 COP (approx 5 USD) for using the chairs, compressor and the remaining facilities.
Kitespot Playa Central (Downtown Beach)
Another very popular kitespot is the main beach in the north, which is usually quite packed with tourists sun bathing, so starting and landing here will sometimes be a bit interesting too, but at least the beach has more depth than Chamey's place. Locals from Lago Calima started offering also kite lessons here during peak season for around $100.000 COP (approx 50 USD).
Despite its popularity (possibly for showing off the tricks to the crowd of tourists and its location) you should be prepared for a mix of boats, jet skis and tourists in the water, it can get crowed from time to time. The water here is fairly choppy and I personally do not recommend this spot for beginners. Nevertheless it is the perfect spot for starting one of the beautiful downwind sessions and it has also a flat water spot close to the hotels.
If you are more into wave riding you have to either ride all the way out to the reef (2-3km) or stick around more in the south of the island, as the outer reef does not protect this region and allows for the swell to come to the shore. The first waves you will usually encounter around Rocky Cay and can pick your preferred spot along the remaining south-east coast. The next larger sandy beach will be in San Luis, further south there is also a shop that offers windsurfing lessons, so I wouldn't be surprised if it is a decent place for wave riding as well. If you feel like a change for once you can also book a boat to Cayo Bolivar (two uninhabited small islands 20km east of San Andres), the boat operator will also happily take your kite equipment.