I've been cycling in Gozo since July, clocking up around 600km according to my GPS (which is pretty good considering the island is barely 15km x 7km). While you should take note of the warnings from Mark and Mr Kriss, it's a superb place to cycle, much more so than Malta and among the best in Europe. It deserves a post of its own but basically the locals don't know how to deal with cycles and are often under the impression that cyclists should always give way to motor vehicles. But if you go off-road that's not an issue and on the roads simply steer well clear of anything with four or more wheels.
You can potentially cycle all year round, though I've found it a bit cold in December as I didn't bring any long sleeved/legged gear. In June-September you'll have to plan to be off the bike and in the shade between 12:00 and 16:00 each day. September/October and May/June are probably best if cycling is your main objective.
Gozo has something for every type of cyclist. My son introduced me to off-road biking here but if you prefer a leisurely pedal between churches and scenic views or to take on steep climb challenges there's plenty to keep you busy. You can see (far too many) of my Gozo cycling photos at http://www.flickr.com/photos/softrax/collections/72157607934219981/.
I'm only aware of one person who does any sort of organised cycling and I don't have any contact details. If you were to say how long you wanted to stay and what you're interested in (serious off-road, scenery, beaches, history, culture, churches) I could suggest an itiniery for you.
The tourist maps are rubbish incidentally. If you plan to go offroad then the Kompass
Bicycles are available for hire for aound €5/day or €25/week. I can't name the place I've used in this topic but there are garages that hire bikes in Victoria and I think both Xlendi and Marsalforn and probably elsewhere. Look for a light bike with full suspension or front suspension and good gel saddle - your backside will thank you for it.
SAFETY WARNING: During the siesta time in summer there's a serious risk of of severe sunburn, heatstroke and dangerous dehydration. That should be obvious but I've seen people who ignored it and it wasn't pretty. Always carry at least 500ml of water for every hour you plan to be out in summer, half that the rest of the year (and isotonic or energy replacement drinks for more than three or four hours). It's fair to say that your safety is in your own hands. Gozo is a lump of rock averaging around 100m high with few beaches so if you go far enough in any direction there's an 80% chance you'll drop off a cliff. They don't do barriers or warnings or any of that safety stuff here so if you can see the sea anywhere and you can't see the ground a few metres ahead, that's probably because there isn't any. Don't let that put you off because as long as you're aware and you use common sense you'll be fine.