I am both a binocular and camera enthusiast.
First tell what you are willing to pay then I or others can give you a recommendation for both camera and binocular.
Only about 200 or less dollars. I just want to be able to differentiate birds at the feeder.
I can probably only afford $400 or less for a camera. I may have to save up to buy something higher priced. I currently have a Panasonic ZS7 with 12x zoom, but it is nowhere near powerful enough. My binoculars are probably very inexpensive, a pair of little Bushnells, 8 x 21.