For ground peppers I just use a variety of containers, I have a few shakers, a woozy bottle, little ziplock baggies.
I'm storing some dried pepper halves in a large ball jar for now, not sure how well they will keep but as long as they aren't exposed to sunlight they should keep all winter.
Some people freeze fresh peppers for use later or for sauces but after freezing cell structure breaks down and they get mushy when heated/defrosted. Keep that in mind.
I may end up drying and vac sealing a bunch of peppers as well for grinding later in winter.
As for fresh peppers they probably keep longer in the fridge than on the counter but I usually have a big bowl of peppers on counter for when i'm cooking.