Welcome to a fellow CNY'er.
I'm west of Syracuse a little ways, and the smoking woods abundant in CNY are maple (sugar & silver), apple (any variety), and cherry (mostly black cherry). Oak is also common.
Chips from the store are certainly the best bet to get started while finding a good source of wood.
As mentioned above, check with tree services. Often they simply chip up what they cut down, but if you let them know you're interested in any of the "smoking" woods, they may give you a call when they have a job cutting one down.
If you Learn what the bark on these trees look like, should you buy firewood, you can come across some nice splits that you can chunk up for smoking too.
I'm fortunate to have a severely overgrown apple tree and cherry tree in my yard. I waited until all the leaves were off the cherry tree (this week) and just cut it back hard, yielding 1/4 face cord of wood for smoking. Once the apple tree is done for the year, I will be getting out the chainsaw again and stocking up on apple as well. With two HUGE silver maples in the back yard, I get decent size branches to chunk up every time we get high winds (which is pretty often lately) The maple branches are already seasoned - most are branches that broke off during an ice storm a few years ago and got hung up in the tree.
Good luck!