If you're on a road bike, the internal loop road around Central Park is known officially as Park Drive, and it's approximately 6 miles around and is closed to cars on weekdays from 10am-3pm and 7pm - 10pm, and all weekend long, from Friday night at 7pm until Monday morning at 6am. From the day before Thanksgiving until the first business day of the new year, the drive is still closed to cars on weekends, but it's open to cars 24 hours a day on weekdays... for Christmas shoppers, I guess.
You can enter the Park Drive at any of about 20 entrances to the park, although some of the bigger cross streets, like 79th or 86th or whatever, don't actually enter into the park, they go sort of under it, so avoid those (see map link, below).
If you're on a mountain bike, be sure to check out the park's "Bridle Path," which is a really fun country-style dirt road that was formerly used for horseback riding but, since the last stable in NYC closed its doors in 2007, it's not used by horses any more, so they don't chase bikes off of it any more. But, even so, if you're on the bridle path and you see a cop, it's a good idea to go the other way, if you know what I'm talkin' about. But don't let that deter you, it's definitely worth riding, and your chances of seeing a cop in there are, like, virtually nil!
Good luck, and the link below is to a pdf map of the park that shows both the loop road and the bridle path, too. It's a runner's map, but don't worry, the colored lines show the loop drive, the dotted line is the bridle path, and the white lines are pedestrian-only footpaths...