Onondaga County's Parks have lots of events right through the holiday week, some organized, some 'do it yourself, on your own schedule.'
Parks Commissioner Bill Lansley says Lights on the Lake, the Onondaga Lake Parkway drive-thru, is open to car traffic every night except New Year's Eve, when they do First Night CNY, a ticketed event (tickets at the 'door'). But, he also says there's a 'free' way to see the lights, which are turned on about 3pm: you can walk thru at dusk, magical in itself.
Beaver Lake has lots of outdoor walks, or check out the trails yourself. If there's snow, there are also snowshoe rentals.
Highland Forest has horsedrawn rides all the way thru New Year's and then weekends til Spring. There's also a new ski trail there, and more wintertime events.
And, Syracuse's zoo has reduced admission in January. There's also a photo contest--download the form and you can get in free.
Onondaga County Parks website or 451-PARK have the details