Orlando Proper and Seminole County have a ton of Bike Trails throughout the area, and Orange County has some as well!
For 'hiking' and nature trails, Wekiva is a great go too, but it can get busy there. I'm more familiar with the bike trails, so if you have any specific questions I'm open there.
Here are some good resources. Check out the maps on each of the sites to get a good perspective.
The City of Orlando trail network consists of over 40 miles and offers a combination of off-street paths, and asphalt and concrete sections.
www.orlando.gov