Hey welcome to the forums!

I live in central NY too! I was vacationing in Cape Cod in Sept too, in Wellfleet! I rode some of the rail trail there- maybe we passed each other!

As to riding in wet weather....
If it's raining hard, you'll want to go very slow and test your brakes often- if they get too wet you will lose some braking power and you don't want to find that out when you are stopping at a busy intersection.

Usually if it's raining lightly or just wet:
Sure, slow down a bit, do test your brakes every few minutes just to be sure.
Make sure your tires are in good shape- not going bald or cracked and old.
Might want to inflate your tires just a bit less than usual to get more gripping surface on the road.
Obviously, wear clothes for the occasion. Aside from waterproof outer jackets- consider merino wool clothing and wool socks to keep you from feeling too cold and soggy.

Also, invest in some lights for your bike in rainy weather. Either permanent head and tail lights, or else a couple of clip on battery run "blinkies" both front and rear, so that cars can see you from far away.
Remember that puddles can hide some nasty potholes, rocks, or other debris- try to avoid going through puddles when you can. Also avoid riding through piles of wet leaves- they are slippery.
Fenders are great to have if you commute to work and ride through rainy weather regularly.