Selecting mature fruit from the tree

[BajaJohn]

Some fruits more than others are at their best when left to mature on the tree. I'm also guessing that some fruit keeps better on the tree than if they are harvested and stored, unless you have special storage facilities.
My question is how to harvest and/or treat fruit for the best eating quality. One issue for me is how to select fruit on the tree that is mature/ready for picking. I'm sure that it depends on the fruit and non-commercial growers have much more leeway for getting that perfect fruit, so I'd like to hear how people do this.
I was never able to pick good apricots from the tree and noted that the best tasting were the ones that had fallen, so I rigged up a cloth to catch the fruit when it fell. I couldn't see/feel any difference between the fruit still on the tree and the fruit I caught in the cloth, but the fallen apricots were way juicier and sweeter. I do the same with my mangos now - in part because because of the numbers and the trees are so high that it would be a major undertaking to check the individual fruits.
Citrus, I used to check with a gentle tug but the mandarins often sounded like I was tearing something inside. I now rotate the fruit 90 degrees to the stem and pull even more gently. That seems to work well with Valencia and Mandarin oranges and grapefruit but doesn't work with Key limes which are at their best when still green, before they yellow and they are far from their best if they are left to fall from the tree.
I also seem to find that the citrus at the top and on the South side of the trees seem to mature first, presumably because of sun exposure.

[BajaJohn]

There seems to be more interest in this topic now that mangos are ripening.

Quality of mangoes will be much better if they are not left on the tree until they naturally drop.  Even if letting color up, which many should do before picking, you still want to pick before it is at the stage where it drops off on its own.

Once picked, let it sit and ripen on a table on your porch or the like where they will be exposed to natural outdoor ambient temperatures.

I can't agree with such a generic statement, even for mangos. I suspect there are many variables including climate, variety, personal preference and possibly even the stage of the harvest and age of the trees.
My favorite crop variety is known here as manzana (apple) which is absolutely the best when left to fall from the tree. It is an old tree and I have no idea what modern variety it corresponds to. They look somewhat like Haden and there is a picture of some early in the season below. They are about twice the size later in the season and ripen a month or two later than my ataulfos (manillas).

My ataulphos are just beginning to ripen now. Even most of the ones falling from the tree still show green and need time to ripen. I note the turpentine taste is stronger in the one that fall from the tree still green. Later in the season the turpentine flavour lessens but the fruit seems a little overripe and more fibrous if left to fall naturally from the tree.

[MarvelMango]

Most places pick mangos way to green. For instance until recently I didn't know keits and kents turned color, I always thought they were green mangos. 
I prefer to let them ripen to with in a day or two on the counter. My experience is mainly with Haden and like mango's as that is what was found in everyones yard when I was growing up in this area. But I noticed when a mango colors up it still has a dull look to it. But when it's very near to being ripe that dull color turns to a waxy shinny look. Thats when I try to pick them.
But as I'm learning more and eating more different varietys, I've learned that this doesn't hold true to all mango's.