We like to cycle in the area of 25 - 30 miles a day, and have had fun renting bikes at WDW, but it is not the type of cycling we usually do. There are some trails around Wildnerness Lodge and a trail from OKW to downtown DIsney, but they are very short.
I have not ridden it, but plan to try the "orange blossom trail" nearby. do a web search and you should find information.
About an hour east of Disney is Cocoa Beach and we like to stay there in the winter to get lots of riding in. There is cycling both on the beach (uninterrputed 6 miles) and bike trails along the causeway and around the area. It is very bicycle friendly with full bike lanes on major roads. We can get several hours of good cycling in very safely there on dedicated trails. Vero Beach also has bike trails heading along A1A both north and south - we followed bike trails literally the whole coast from Cocoa beach to Vero Beach.
I wish WDW would do more to help cyclists - they have the land. I would not venture out onto the roads - too many multi-land with heavy traffic.