I lived in Clinton / New Hartford for a couple of years and, honestly, the trail selection around Utica / Rome isn't all that great. I spent most of my time driving all over NY for long rides and races.
I spent a lot of time in the Sherril Brook Park in New Hartford. The size is pretty deceiving and it goes back quite a ways. There are some great "hidden" trails back there. You've gotta spend some time exploring all of the trails before you find a loop that challenges you. I had a great 5 mile loop that could almost challenge any trail around me now (Rochester area, Tryon, Spencer, Letchworth, Highland Forest, etc).
From Utica, take the Arterial (12) south to New Hartford and take 12 at the first big light to the left towards the New Hartford village. Take the first right (big light again, 12 again) which takes you up a hill. Go about 1/2 - 1 mile and the park is on your right. Or search Mapquest for New Hartford, NY and zoom in one or two clicks. You'll see the big green patch to the west of New Hartford village.
Highland Forest in Fabius is about 45-60 minutes away and great for a weekend daytrip. McCauley Mountain in Old Forge is fun. You just have to park outside of the gates and ride in. I'm really not sure if they allow riding, though... I just do it anyway
http://bikeloft.com/site/page.cfm?PageID=37 has a decent list of the better rides in Central NY.
Good luck!
- Jeremy -