I can't seem to get it right, sometimes I put certain fruits in the fridge and they taste horrid, ao I keep them out and they go off

I've recently been getting a box of fruit delivered fortnightly from Aussie Farmers Direct and I find i'm throwing so much out because I don't know how to store it properly! Here's some of the stuff I get in the box...

Grapes - wash then fridge (they're eaten in a day or two so always fine)
Oranges - Sometimes they are ready to eat, sometimes they look like lemons - when do I put them in the fridge if ever??
Lemons and limes - I put straight in fridge
Pears - sometimes they are hard so I leave them out is that right? But sometimes they don't soften
Plums - straight in fridge
Kiwi - fridge
Apples - fridge
Bananas - cupboard, but if there are alot can I put them in the fridge to extend their life? I did that the other day and they all turned black
Necterines - Stopped getting those now but what to do with them?
Peaches - Not getting them anymore either but would still like to know

When you don't put fruit in the fridge, where do you keep it? I always like the look of an overflowing fruit bowl on the counter but should it be kept in the dark??

TIA