Depending upon your location there are a lot of great places. Shawnee Mission Park has some trails set up for Mountain bikes, Cyclocross bikes, and for hiking if you don't want to venture too far out of Johnson County.
Douglas County Lake has some hiking trail, but really the LMBC's River Trail in North Lawrence and Clinton Lake's extensive trail system are the best in Douglas county. The river trail is 10+ miles of fairly fast ridable and run-able trail right along the Kansas River. Clinton Lake is a lot more technical and challenging, and has trails ranging from easy to hard.
Perry lake has a lot of trail too, but I haven't been out there.
There's also Wilson Lake and the trail at the Governor's Mansion.
There's a load of trail in Kansas...
Justin LMBC President