I keep onions and garlic out because I don't want their flavor influencing the flavor of things in the refrigerator. I keep potatoes out because...well, who has room for 10 pounds of potatoes in their refrigerator?? I also keep things like avocados and tomatoes out so they can ripen. But, if I can't get to them right away, I transfer them to the refrigerator so they don't get overripe.
Every other vegetable goes in the refrigerator immediately so they can keep longer.
But, I almost always keep fruits out. This sometimes changes when we get into the hot days of summer, and fruit flies start to show up.