Malapascua & MoalBoal
Diving in Cebu is possible year-round. The best period however is between December & May. The weather is nice, and the visibility is great. Cebu has many great diving scenes including MoalBoal's Sardine run and the thresher sharks of Malapascua.
There are also some great liveaboards departing from Cebu going around the island visiting all the best spots.
Malapascua is a remote island all the way to the north of Cebu. The island I feel is divided into 2 parts: The tourist part, and the non-tourist part. From the beach, you will end up in the tourist part which includes shops, hotels, and dive centers.
If you talk a walk onto the island you will visit the people as they live there and with luck witness some party or festival. The island is gorgeous and easily walkable.
Diving is amazing! The signature dive here is with the thresher sharks. The dive will start early(!) in the morning just before sunrise and will take you to approximately 30 meters deep where you can sit down on the bottom floor and watch the sharks come up to the cleaning station in the morning. The site is not a guarantee for thresher sharks though. It took me 2 dives to see one but was well worth it.
Moalboal is a lively little town in the southeast part of Cebu. The town is renowned for its Sardine run: A shore dive right off the beach, dropping to 30 meters or so and here you will witness millions of sardines! They will black out the sun making this dive almost feel like a night dive. For this reason, only you should visit, but the surrounding waters provide a lot of interesting dive sites with amazing reefs and even some wrecks. Night diving is available as well.
How to get there?
Malapascua: To get there can be quite a trip since you need to fly to Cebu, get a taxi (which you can probably share), and will take you about 3-4 hours to the jetty in New Maya to Malapascua. Then it's a short ferry to Malapascua and well worth it (100 pesos).
Moalboal: To go to Moalboal, you can take the bus that says Bato via Barili or Moalboal from Cebu City. If you want to go to Panangsam or white beach you will have to take a motorcycle.