Generally speaking, you're going to need the following items:
Backpack
Shelter (tent)
Sleeping Bag (make sure the comfort rating is good. Usually the limit rating is what's advertised)
Sleeping Pad (make sure it has a high R value to keep you warm)
A way to filter water
A way to boil water (Stove, lighter, cook pot, etc.)
Something to carry your water in
A source of light (headlamp)
BooBoo Kit (bandaids, pain meds, etc.)
Supplies to do the do (TP, Trowel, etc.)
I have a google sheet that I've been putting together which has a more detailed list of gear. You can check that out here.
The Adirondacks has rules about storing food. They require a bear canister to be used when in the High Peaks. So, if we're going to Mt Colden, then this is a must, but I would highly advise using a bare canister anyway.
There's a specific bear canister that the park wants you to use as there was a bear a while back that figured out how to open the other type of bear canister.
You need to get the type with the locking pins, not one with a twist close top.
The one I have is from Backpackers Cache. Here's an Amazon link.
They are also available for rent from the park.
Here's a few example photos of what I typically bring on a backpacking trip.